Arcs Intersecting at Most Once on the 4-Punctured Sphere

Paul Tee

Abstract

We classify maximal systems of arcs which intersect at most once on the
4-punctured sphere.

###### Contents

  1. 1 Acknowledgements
  2. 2 Introduction
  3. 3 Classifying Saturated 0-Systems
    1. 3.1 Tree $T_{a}$
    2. 3.2 Tree $T_{b}$
  4. 4 Properties of 1-systems
  5. 5 Classifying Maximal 1-Systems
    1. 5.1 $|\mathbb{J}|=3$
    2. 5.2 $|\mathbb{J}|=2$
    3. 5.3 $|\mathbb{J}|=1$
    4. 5.4 $|\mathbb{J}|=0$

##  1 Acknowledgements

First and foremost, I would like to thank my advisor Piotr Przytycki for not
only supporting me academically but also for helping me adjust to a new life
in a difficult time. Thank you to Denali Relles for noticing an extra maximal
1-system in an earlier version of this paper. Thank you to my wonderful
girlfriend who has been my rock throughout these two years. Thank you to
parents who have been there for me throughout this journey of life. Lastly,
thank you to my friends, especially to those in and around the reading room.

##  2 Introduction

In this paper, we classify maximal systems of arcs which intersect at most
once on the 4-punctured sphere. We begin with a survey of the relevant
definitions following [2].

Let $S$ be a surface of finite type with punctures and Euler characteristic
$\chi<0$. In certain natural situations, we choose to call punctures vertices.
An arc on $S$ is a map from $(0,1)$ to $S$ that is proper. A proper map
induces a map between topological ends of spaces, and in this sense each
endpoint of $(0,1)$ is sent to a puncture of $S$. We will say that the arc is
between these punctures. An arc is simple if it is an embedding. In that case
we can and will identify the arc with its image in $S$. An arc $\alpha$ is
essential if it cannot be homotoped into a puncture in the sense that there is
no proper map $(0,1)\times[0,1]\to S$ which, restricted to $(0,1)\times 0$,
equals $\alpha$. All arcs in this paper are simple and essential. An arc is a
loop if it starts and ends at the same puncture. In this case, we say the loop
is based at the puncture. A homotopy between arcs $\alpha$ and $\beta$ is a
proper map $(0,1)\times[0,1]\to S$ which, restricted to $(0,1)\times 0$ equals
$\alpha$, and restricted to $(0,1)\times 1$ equals $\beta$. In particular,
$\alpha$ and $\beta$ start at the same puncture and end at the same puncture.

We say that arcs $\alpha$ and $\beta$ are in minimal position, if the number
of intersection points $|\alpha\cap\beta|$ cannot be decreased by a homotopy.
A bigon (respectively, half-bigon) between arcs $\alpha$ and $\beta$ is an
embedded closed disc $B\subset S$ (respectively, properly embedded half-disc
$B=[0,1]\times[0,1)\subset S$) such that $B\cap(\alpha\cup\beta)=\partial B$
and both $\partial B\cap\alpha$ and $\partial B\cap\beta$ are connected. It is
a well known fact, we call the bigon criterion for arcs, that $\alpha$ and
$\beta$ are in minimal position if and only if they are transverse and there
is no bigon or half-bigon between them.

We define a $k$-system $\mathbb{A}$ on a surface $S$ to be a collection of
pairwise nonhomotopic arcs such that any two arcs intersect at most $k$ times.
Note that a $0$-system is a set of disjoint arcs. We define k-systems
$\mathbb{A},\mathbb{B}$ to be equivalent if for every arc
$\alpha\in\mathbb{A}$ there exists an arc $\beta\in\mathbb{B}$ such that
$\alpha\simeq\beta$, where $\simeq$ denotes homotopy. We denote this by
$\mathbb{A}\simeq\mathbb{B}$. Define the degree of $\mathbb{A}$ to be
$\deg(\mathbb{A}):=(\deg p_{i})_{i\in\mathcal{P}}$ where $\mathcal{P}$ denotes
the set up punctures equipped with a total order. If
$\mathbb{A}\simeq\mathbb{B}$, then up to reordering,
$\deg(\mathbb{A})=\deg(\mathbb{B})$.

We say $\mathbb{A}$ is saturated if for any arc $\sigma$,
$\\{\sigma\\}\cup\mathbb{A}$ is no longer a $k$-system. Theorem 1.5 of [1]
implies for a fixed surface $S$, the cardinality of any saturated $k$-system
is bounded by a constant that depends only on $S$ and $k$. Thus, we say a
saturated $\mathbb{A}$ is a maximal $k$-system if for any saturated
$\mathbb{B}$, we have

$|\mathbb{B}|\leq|\mathbb{A}|.$

Consider a $k$-system $\mathbb{A}$ where the arcs are pairwise in minimal
position. There is a canonical subset, the disjoint subset,
$\mathbb{J}\subset\mathbb{A}$ consisting of arcs which are disjoint all other
arcs in $\mathbb{A}$. In other words,

$\mathbb{J}:=\\{\sigma\in\mathbb{A}|\sigma\cap\tau=\emptyset,\forall\tau\in\mathbb{A}-\\{\sigma\\}\\}$

######  Remark 2.1.

Any saturated 0-system on $S$ has cardinality $3|\chi|$.

###### Proof.

Choose a hyperbolic metric for $S$, and then by the Gauss–Bonnet Theorem, the
area of $S$ is $2\pi|\chi|$. Any saturated system forms an ideal triangulation
of $S$. As any ideal triangle has area $\pi$, we have $2|\chi|$ triangles, so
$6|\chi|$ edges. Every edge is counted twice in a triangulation, so we have
$3|\chi|$ arcs. ∎

The following provides an analogous statement of Remark 2.1 for 1-systems.

######  Theorem 2.2.

[1, Theorem 1.2] Any maximal 1-system on $S$ has cardinality
$2|\chi|(|\chi|+1)$.

Throughout, denote $\Sigma$ as the four-punctured sphere, and let
$\overline{\Sigma}$ denote the compactification of $\Sigma$ by points labelled
by $\mathcal{P}=(a,b,c,d)$ corresponding to the punctures. We denote our
working representation of $\Sigma$ (equivalently, $\overline{\Sigma}$) as the
3-punctured disk $D$ with boundary $d$, identified as shown in Figure 1. Then,
each arc ending (or starting) at $d$ can be homotoped to an arc that converges
to a point on $\partial D$, and we will be only considering such arcs. Note
that homotopic arcs ending at $d$ might converge to different points of
$\partial D$.

Figure 1: $\Sigma$ as the 3-punctured open disk

Using this representation, we state our two main results. Our classification
is taken with respect to self-homeomorphisms of $\Sigma$ and equivalence of
$\mathbb{A}$.

######  Lemma 2.3.

There are six saturated 0-systems on $\Sigma$, as show in Figure 2.

Figure 2: Saturated 0-systems labelled (i)-(vi)

######  Theorem 2.4.

There are nine maximal 1-systems on $\Sigma$, as shown in Figure 3.

Figure 3: Maximal 1-systems labelled $J_{3a}-J_{0}$

Note that for each maximal 1-system the black arcs are exactly $\mathbb{J}$.
The naming convention is as follows: the number denotes the number of arcs in
$|\mathbb{J}|$, and for a fixed $|\mathbb{J}|$, the letters (if they exist)
index the distinct maximal 1-systems.

Organization: In Section 3 (resp. Section 5), we use Remark 2.1 (resp. Theorem
2.2) to classify saturated 0-systems (resp. maximal 1-systems) on $\Sigma$. In
Section 3, we first show any saturated 0-system must contain an embedded tree,
and the two subsections deal with each case separately. In Section 4, we
discuss several technical lemmas we need through the rest of the paper, before
culminating in showing $|\mathbb{J}|\leq 3$. This section can be used as
reference material for Section 5. In Section 5, the subsections 5.1, 5.2, 5.3,
5.4 describe the $|\mathbb{J}|=$ 3, 2, 1 and 0 cases respectively.

##  3 Classifying Saturated 0-Systems

Let $\mathbb{A}$ be a saturated $0$-system on $\Sigma$. As $\chi(\Sigma)=-2$,
we have $|\mathbb{A}|=6$ by Remark 2.1. We begin by observing that
$\mathbb{A}$ defines a embedded graph
$\Gamma=\Gamma_{\mathbb{A}}\subset\overline{\Sigma}$ with vertex set
$\mathcal{P}$ and edge set $\mathbb{A}$.

######  Lemma 3.1.

If $\mathbb{A}$ is saturated and $\Gamma$ is as defined above, then $\Gamma$
is connected.

###### Proof.

Indeed, suppose $\Gamma$ is not connected. Take any arc
$\sigma\subset\overline{\Sigma}-\Gamma$ between any two connected components
which misses $\Gamma$ except at endpoints. Note that
$\\{\sigma\\}\cup\mathbb{A}$ defines a $0$-system in $\Sigma$, giving the
desired contradiction. ∎

By Lemma 3.1, we can find a tree $T\subset\Gamma$ with four vertices. As there
are two isomorphism types of maximal trees on four vertices, the two trees
$T_{a},T_{b}$ in Figure 4 illustrate the two different ways of embedding a
tree into $\overline{\Sigma}$, and this embedding is unique up to a
homeomorphism of $\overline{\Sigma}$.

Figure 4: Embedded maximal trees $T_{a},T_{b}$

We complete $T$ to a saturated 0-system if we can find three disjoint,
nonhomotopic arcs in $\overline{\Sigma}-T$. Begin by observing that cutting
along $T$ gives an ideal hexagon in either case. To complete $T$ to a
saturated 0-system, we have three configurations of arcs as seen in Figure 5.
The remaining work is to put the labels corresponding to $T_{a},T_{b}$ into
the hexagon.

Figure 5: $\Delta,\epsilon,$ N configurations

###  3.1 Tree $T_{a}$

Note there is an action of $\mathbb{Z}/2$ on this graph, which interchanges
$a$ with $d$ and $b$ with $c$. We can extend elements of $\mathbb{Z}/2$ to
homeomorphisms of $\overline{\Sigma}$ which preserve $T_{a}$. We classify the
configurations of Figure 5 up to these homeomorphisms.

  1. $\Delta$:

Up to an action of $\mathbb{Z}/2$, there is one $\Delta$ configuration,
illustrated in Figure 6.

Figure 6: Homeomorphic to (vi) in Figure 2

  2. $\epsilon$:

Up to an action of $\mathbb{Z}/2$, there are two $\epsilon$ configurations,
illustrated in Figure 7.

Figure 7: Homeomorphic to (i), (vi) in Figure 2

  3. N:

Up to an action of $\mathbb{Z}/2$, there are three $N$ configurations,
illustrated in Figure 8.

Figure 8: Homeomorphic to (i), (ii), (v) in Figure 2

###  3.2 Tree $T_{b}$

Note that for $V=\\{a,c,d\\}$, we have an action of the symmetry group $S_{V}$
on this graph. We can extend elements of $S_{V}$ to homeomorphisms of
$\overline{\Sigma}$ which preserve $T_{b}$. We classify the configurations of
Figure 5 up to these homeomorphisms.

  1. $\Delta$:

Up to an action of $S_{V}$, there are two $\Delta$ configurations, illustrated
in Figure 9.

Figure 9: Homeomorphic to (ii), (iii) in Figure 2

  2. $\epsilon$:

Up to an action of $S_{V}$, there are two $\epsilon$ configurations,
illustrated in Figure 10.

Figure 10: Homeomorphic to (iv), (i) in Figure 2

  3. N:

Up to an action of $S_{V}$, there is one $N$ configuration, illustrated in
Figure 11.

Figure 11: Homeomorphic to (vi) in Figure 2

We can distinguish systems (i)-(vi) by consideration of the set of the vertex
degrees of $\Gamma$, which is easily seen to be an invariant of $\mathbb{A}$.
Thus, all saturated 0-systems on $\Sigma$ are given by Figure 2 up to
homeomorphisms and equivalence.

##  4 Properties of 1-systems

Let $\mathbb{A}$ be a maximal $1$-system on $\Sigma$. As $\chi(\Sigma)=-2$, we
have $|\mathbb{A}|=12$ by Theorem 2.2.

######  Lemma 4.1.

Consider a loop $\ell$ based at puncture $p$ which encloses a once-punctured
disk $D_{pq}$ with puncture $q$. Let $\sigma_{pq}\subset D_{pq}$ be an arc
between $p$ and $q$. For any maximal 1-system $\mathbb{A}$ such that
$\ell\in\mathbb{A}$, then necessarily $\sigma_{pq}\in\mathbb{J}$.

###### Proof.

For any $\alpha\in\mathbb{A}$, $\alpha$ must satisfy exactly one of the two
conditions:

  1. 1.

$\alpha$ can be homotoped to an arc inside $\Sigma-D_{pq}$.

  2. 2.

$\alpha$ has an endpoint at $q$ and connected intersection with $D_{pq}$.

These two conditions follow from the bigon criterion for arcs. The lemma
follows as each arc in condition 2 is homotopic to or disjoint from
$\sigma_{pq}$. ∎

######  Lemma 4.2.

Let $\mathcal{D}_{n}$ be a once-punctured $n$-gon for $n=1,2,3,4$ with
puncture $p_{-\infty}$, and $\mathbb{A}=\mathbb{A}_{n}$ a maximal 1-system on
$\mathcal{D}_{n}$. Then $\mathcal{D}_{n}$ contains at most $1,3,6,10$ arcs
respectively.

###### Proof.

The 1,2-gon cases are obvious. For $\mathcal{D}_{3}$, note that there are 3
loops and 6 arcs which are not loops - 3 joining vertices and 3 joining a
vertex with a puncture.

Figure 12: Correspondence between arcs joining distinct vertices and loops

However, not all configurations of arcs are possible. In Figure 12, the blue
loop excludes the following orange arc, and this type of restriction is the
only possible one. Consequently, there are at most 6 arcs in $\mathbb{A}$. In
fact, see Figure 13 for an example.

Figure 13: A maximal 1-system on $\mathcal{D}_{3}$

For $\mathcal{D}_{4}$, we count 4 loops, and 12 arcs which are not loops. This
comes from 4 arcs joining a vertex to the puncture, 4 arcs joining adjacent
vertices which we call _adjacent arcs_ , and 4 arcs which join opposite
vertices. The restriction in $\mathcal{D}_{3}$ generalizes to the following
statement: Suppose first there is a loop $\ell\in\mathbb{A}$. Apart from other
loops, we have exactly three arcs, illustrated in Figure 14, which intersect l
more than once and hence are excluded from $\mathbb{A}$. This shows
$|\mathbb{A}|\leq 10$. Observe also, we can have at most two adjacent arcs in
any 1-system, and these two adjacent arcs must share a common endpoint.

Figure 14: Arcs excluded by a loop

If we have no loops, it is clear the constraints discussed above force any
1-system to contain at most ten arcs. ∎

Finally, consider $\Omega=S^{1}\times[0,1]$, where each boundary component is
given a cell decomposition of two vertices and two edges. In the following
discussion, we only consider arcs which are not loops. Label
$S^{1}\times\\{0\\}$ vertices $a,d$ and $S^{1}\times\\{1\\}$ vertices $b,c$.
Let $\alpha_{0}=\\{a\\}\times[0,1]$ be the “vertical arc” between $a,b$. For
$n\in\mathbb{Z}$, denote $\alpha_{n}$ as image of $\alpha_{0}$ under $n$ Dehn
twists about $S^{1}\times\\{\frac{1}{2}\\}$. Observe

$|\alpha_{n}\cap\alpha_{m}|=|n-m|-1$

Similarly, define $\delta_{0}=\\{d\\}\times[0,1]$, and $\delta_{n}$ the image
of $\delta_{0}$ under $n$ Dehn twists about $S^{1}\times\\{\frac{1}{2}\\}$,
and a similar observation applies.

We generalize this observation. Let $X$ denote the set of homotopy classes of
arcs between the two boundary components which start and end at a vertex.
Thus, $X=X_{a}^{b}\sqcup X_{a}^{c}\sqcup X_{d}^{b}\sqcup X_{d}^{c}$, where
$X_{a}^{b}$ denotes the set of homotopy classes of arcs between $a$ and $b$,
and so on. Define a map

$\tau:X_{a}^{b}\sqcup X_{d}^{c}\to\mathbb{Z}$
$\tau:\alpha_{n},\delta_{n}\mapsto n$

For $\sigma\in X_{a}^{c}\sqcup X_{d}^{b}$, choose a representative which lies
in the disk bounded by $\alpha_{n}\cup\alpha_{n+1}$ and
$\delta_{n}\cup\delta_{n+1}$ for a unique $n\in\mathbb{Z}$.

Figure 15: $\sigma$ in the region bounded by $\alpha_{0},\alpha_{1}$ (left)
and $\delta_{0}$, $\delta_{1}$ (right)

Then the formula

$\tau(\sigma)=n+\frac{1}{2}$

defines a map $\tau:X\to\frac{1}{2}\mathbb{Z}$ which we call the twisting
number. By induction, it is easy to prove that $\tau$ computes the number of
intersections of two arcs in minimal position:

######  Lemma 4.3.

Let $\sigma_{1},\sigma_{2}\in X$, and let $\eta$ denote the number of common
endpoints between $\sigma_{1}$ and $\sigma_{2}$. Then,

$|\sigma_{1}\cap\sigma_{2}|=|\tau(\sigma_{1})-\tau(\sigma_{2})|-\frac{\eta}{2}$

Figure 16: Examples of $\frac{1}{2}\mathbb{Z}_{a}$ (green) and
$\frac{1}{2}\mathbb{Z}_{d}$ (orange)

We next introduce two copies of $\frac{1}{2}\mathbb{Z}$, labelled
$\frac{1}{2}\mathbb{Z}_{a}$ and $\frac{1}{2}\mathbb{Z}_{d}$ corresponding via
$\tau$ to $X_{a}^{b}\sqcup X_{a}^{c}$ and to $X_{d}^{b}\sqcup X_{d}^{c}$,
respectively. Some examples are illustrated in Figure 16.

######  Remark 4.4.

By Lemma 4.3 for $\sigma_{1},\sigma_{2}\in\frac{1}{2}\mathbb{Z}_{a}$, we have
$|\tau(\sigma_{1})-\tau(\sigma_{2})|\leq 2$, so in particular there are most
five arcs in $\frac{1}{2}\mathbb{Z}_{a}$. Furthermore, for
$\sigma_{1}\in\frac{1}{2}\mathbb{Z}_{a}$ and
$\sigma_{2}\in\frac{1}{2}\mathbb{Z}_{d}$, we have
$|\tau(\sigma_{1})-\tau(\sigma_{2})|\leq\frac{3}{2}$

Define the systems $\mathbb{F},\mathbb{G}$ as shown in Figure 17. Since
$\deg(\mathbb{F})=(6,6,4,4)$ and $\deg(\mathbb{G})=(5,5,5,5)$,
$\mathbb{F}\not\simeq\mathbb{G}$. By Remark 4.4, we have the following:

Figure 17: $\mathbb{F},\mathbb{G}$ maximal 1-systems not containing loops

######  Corollary 4.5.

$\mathbb{F}$ and $\mathbb{G}$, as shown in Figure 17, are the two maximal
1-systems on $\Omega$ which do not contain loops.

The following proposition shows that any maximal 1-system on $\Sigma$ must
have nonseparating disjoint subset.

######  Proposition 4.6.

Let $\mathbb{A}$ be a maximal 1-system on $\Sigma$, in which all arcs are in
pairwise minimal position. Then the disjoint subset $\mathbb{J}$ is
nonseparating, that is, $\Sigma-\mathbb{J}$ is connected.

###### Proof.

Suppose otherwise, so let $\mathbb{J}$ be the disjoint subset of maximal
1-system $\mathbb{A}$, and let $\mathbb{J}$ be separating. For each arc
$\alpha\in\mathbb{A}-\mathbb{J}$, notice that $\alpha$ is contained in a
connected component of $\Sigma-\mathbb{J}$. Let
$\mathbb{C}\subset\Gamma_{\mathbb{J}}$ be a cycle. As we have four punctures,
we have $|\mathbb{C}|\leq 4$.

Figure 18: Cut $\mathcal{W}$ along $\sigma$ (black)

First, assume $|\mathbb{C}|=1$, so $\mathbb{C}=\ell_{p}$, a loop based at $p$.
$\ell_{p}$ divides $\Sigma$ into two components, one component with one
puncture and the other component with two punctures. By Lemma 4.2, the
component with one puncture has at most 1 arc. For the other component
$\mathcal{W}$, consider the unique class of arc between the two punctures. We
can choose a representative $\sigma$ such that any arc which is not a loop can
be taken to miss this $\sigma$. Cutting along this arc, we are reduced down to
Remark 4.4, where we maximize the number of arcs between punctures $a,b,c$.
Thus, we have $5$ arcs which are not loops in $\mathcal{W}-\sigma$, shown by
the orange arcs in Figure 18. It is clear we can add at most 1 loop in
$\mathcal{W}-\sigma$ based at the left puncture. This gives a total of at most
$9$ arcs in $\mathbb{A}$, a contradiction.

If $|\mathbb{C}|=2$, then $\Sigma-\mathbb{C}$ must be two once-punctured
bigons. By Lemma 4.2, each punctured bigon has at most $3$ arcs. This gives a
total of at most $8$ arcs in $\mathbb{A}$, a contradiction.

If $|\mathbb{C}|=3$, then $\Sigma-\mathbb{C}$ must be a triangle and a once-
punctured triangle. By Lemma 4.2, the punctured triangle has at most $6$ arcs.
This gives a total of at most $9$ arcs in $\mathbb{A}$, a contradiction.

If $|\mathbb{C}|=4$, then $\Sigma-\mathbb{C}$ must be two squares. As each
square contains at most 2 arcs, we have at most 8 arcs if
$\Gamma_{\mathbb{J}}$ contains a cycle of length $4$. This gives the desired
contradiction.

∎

For the disjoint subset $\mathbb{J}\subset\mathbb{A}$, let
$\Gamma_{\mathbb{J}}$ the induced embedded graph with vertex set $\mathcal{P}$
and edge set $\mathbb{J}$. Any loops or cycles in $\Gamma_{\mathbb{J}}$ will
separate $\Sigma$. Combining with Lemma 4.6, we see the following corollary.

######  Corollary 4.7.

$|\mathbb{J}|\leq 3$.

##  5 Classifying Maximal 1-Systems

###  5.1 $|\mathbb{J}|=3$

######  Proposition 5.1.

There are two maximal 1-systems on $\Sigma$ for $|\mathbb{J}|=3$, shown in
Figure 19.

As $\mathbb{J}$ is a tree, $\mathbb{J}$ must be either $T_{a}$ or $T_{b}$ from
Section 3. In either case, $\Sigma-\Gamma_{\mathbb{J}}$ is a hexagon, and we
need to find 9 arcs joining two non-adjacent vertices of a hexagon. Any two
pairs of vertices must be used at most once by the homotopy condition. This
forces us to take the complete graph on $6$ vertices. The two 1-systems, each
corresponding to the following graphs in Figure 19. These form distinct
1-systems, as they have nonhomeomorphic $\Gamma_{\mathbb{J}}$.

Figure 19: $J_{3a},J_{3b}$ corresponding to $T_{a}$, $T_{b}$

###  5.2 $|\mathbb{J}|=2$

######  Proposition 5.2.

There are five maximal 1-systems on $\Sigma$ for $|\mathbb{J}|=2$. Two are
shown in Figure 20, two are shown in Figure 21, and one is shown in Figure 22.

If $\mathbb{J}$ contained 2 arcs, the distinguishing factor only is if the
induced graph is connected or not. If so, then $\Sigma-\Gamma_{\mathbb{J}}$ is
a once-punctured square. If not, then $\Sigma-\Gamma_{\mathbb{J}}$ is an
annulus with two vertices and two edges for each boundary component. The two
possibilities correspond to $P_{4}$ in Lemma 4.2, and $\Omega$ in Corollary
4.5. In both cases, we aim to find $10$ arcs in $\Sigma-\Gamma_{\mathbb{J}}$.

Figure 20: $J_{2a},J_{2b}$ on $\mathcal{D}_{4}$ for connected
$\Gamma_{\mathbb{J}}$

Assume $\Gamma_{\mathbb{J}}$ is connected, and we see that we have two cases
in Figure 20. By Lemma 4.1, we have no loops in $\mathbb{A}$. After adding in
labels, We can distinguish the two systems by considering degrees:
$\deg(J_{2a})=(6,10,4,4)$ and $\deg(J_{2b})=(5,10,5,4)$. Thus, for connected
$\Gamma_{\mathbb{J}}$, Figure 20 yields two maximal 1-systems.

Now assume $\Gamma_{\mathbb{J}}$ is disconnected. This brings us to the
$\Omega$, and all that remains is to extend $\mathbb{F}$ and $\mathbb{G}$ in
Corollary 4.5 to maximal 1-systems by adding in loops.

Adding loops to the $\mathbb{F}$ system yields Figure 21. All but two systems
are distinguishable by degree considerations. The systems on the bottom left
and top right are equivalent since one is obtained under an inversion in the
core circle of $\mathbb{A}$. Thus, we have 3 distinct 1-systems corresponding
to $\mathbb{F}$.

Figure 21: $J_{2c},J_{2d}$ extended from $\mathbb{F}$

Adding loops to the $\mathbb{G}$ system yields Figure 22. Note we cut down on
half the cases using symmetry. Furthermore, the half Dehn-twist around the
inside boundary circle of $\mathbb{A}$ maps the left figure to the right one
reflected in the vertical symmetry axis of $\mathbb{A}$. This yields the
unique maximal 1-system corresponding to $\mathbb{G}$.

Figure 22: $J_{2e}$ extended from $\mathbb{G}$

###  5.3 $|\mathbb{J}|=1$

######  Proposition 5.3.

There is one maximal 1-system on $\Sigma$ for $|\mathbb{J}|=1$, shown in
Figure 25.

Without loss of generality, let $\mathbb{J}$ be an arc between punctures
$a,d$, so $\Sigma-\Gamma_{\mathbb{J}}$ is a twice punctured disk. We begin by
discussing arcs which are not loops.

We begin by defining an invariant on homotopy classes of arcs between $a,d$
which we call $\tau^{*}$, and investigate how this relates to $\tau$. We state
a characterization of arcs between $a,d$, following the classification of arcs
on a three-punctured sphere.

######  Remark 5.4.

Let $\sigma$ be an arc between $a,d$, and
$\mathcal{L}\subset\Sigma-\Gamma_{\mathbb{J}}$ a small disk surrounding the
two punctures. Then, every homotopy class of arcs between $a,d$ is obtained by
half-Dehn twists of $\sigma$ about $\partial\mathcal{L}$.

Figure 23: Examples of $\tau^{*}=\frac{1}{4},\frac{3}{4}$

Observe each $\sigma$ between $a,d$ divides the $\Sigma-\Gamma_{\mathbb{J}}$
into two once-punctured bigons. Thus, we get four unique (homotopy classes of)
arcs $\gamma_{i}$, where each arc has one endpoint in $\\{a,d\\}$ and the
other in $\\{b,c\\}$, and is disjoint from $\sigma$. Figure 23 illustrates
this where the $\sigma$’s are the green arcs, and the $\gamma_{i}$’s are the
orange ones. Since there is a unique homotopy class of arcs between $b,c$,
denoted $\phi$, every arc with an endpoint in $\\{a,d\\}$ and the other in
$\\{b,c\\}$ can be characterized by its twisting number $\tau$ in
$(\Sigma-\Gamma_{\mathcal{J}})-\phi$. This accounts for the orange numbers.
Finally, we define

$\tau^{*}(\sigma)=\frac{1}{4}\sum_{i}\tau(\gamma_{i})$

as shown by the green numbers. Note that the image of $\tau^{*}$ is
$\frac{1}{2}\mathbb{Z}+\frac{1}{4}$. In the same spirit as Lemma 4.3, by
induction $\tau$ and $\tau^{*}$ completely determine the number of
intersections of two arcs in minimal position.

######  Lemma 5.5.

If $\sigma_{1},\sigma_{2}$ are two arcs between $a,d$, then

$|\sigma_{1}\cap\sigma_{2}|=2|\tau^{*}(\sigma_{1})-\tau^{*}(\sigma_{2})|+1$

We immediately have as a corollary an upper bound on the number of arcs
between $a,d$.

######  Corollary 5.6.

In any 1-system on $\Sigma-\Gamma_{\mathbb{J}}$, there at at most two arcs
between $a,d$.

######  Lemma 5.7.

If $\sigma$ is an arc between $a,d$ and $\gamma$ is an arc with one endpoint
in $\\{a,d\\}$ and the other in $\\{b,c\\}$, then

$|\sigma\cap\gamma|=2|\tau^{*}(\sigma)-\tau(\gamma)|-\frac{1}{2}$

We classify maximal 1-systems by cases using Corollary 5.6. Suppose we had two
arcs $\sigma_{1},\sigma_{2}$ between $a,d$. Up to a homeomorphism, we can
assume $\tau^{*}(\sigma_{1})=\frac{1}{4}$ and
$\tau^{*}(\sigma_{2})=\frac{3}{4}$. By Lemma 5.7, the permissible values of
$\tau$ are $\tau=0,\frac{1}{2},1$. We illustrate this in Figure 24.

Figure 24: All permissible arcs given $\sigma_{1},\sigma_{2}$

This gives a maximum of: 2 arcs between $a,d$; 6 arcs between $\\{a,d\\}$ and
$\\{b,c\\}$; $1$ arc between $b,c$; $1$ arc in $\mathbb{J}$. As we can have at
most $1$ loop, this brings us to a maximum of 11 arcs. Thus, no such maximal
1-system can exist.

Now assume there is one arc $\sigma$ between $a,d$. Up to a homeomorphism, we
can assume $\tau^{*}(\sigma)=\frac{1}{4}$. By Lemma 5.7, the permissible
values of $\tau$ are $\tau=0,\pm\frac{1}{2},1$. We illustrate this in Figure
25.

Figure 25: $J_{1}$ given $\sigma$ between $a,d$

This gives a maximum of: 1 arc between $a,d$; 8 arcs between $\\{a,d\\}$ and
$\\{b,c\\}$; $1$ arc between $b,c$; $1$ arc in $\mathbb{J}$. As we can have at
most $1$ loop, this brings us to a maximum of 12 arcs. Adding in any loop, we
get the maximal 1-system equivalent to Figure 25.

Now assume there are no arcs between $a,d$. Then $\phi\in\mathbb{J}$,
contradicting $|\mathbb{J}|=1$. Thus, no such maximal 1-system can exist.

###  5.4 $|\mathbb{J}|=0$

######  Proposition 5.8.

There is one maximal 1-system on $\Sigma$ for $|\mathbb{J}|=0$, shown in
Figure 30.

By Lemma 4.1, we observe that no loops can appear in the $|\mathbb{J}|=0$
case. Thus, any arcs mentioned will be understood to be arcs which are not
loops. We begin by proving a slightly stronger statement of Theorem 1.7 of [1]
in the case of the four-punctured sphere.

Let $\mathcal{P}$ denote the four punctures. Let
$\mathcal{Q},\mathcal{Q}^{*}\subset\mathcal{P}$ each be a pair of distinct
punctures such that $\mathcal{Q}\cap\mathcal{Q}^{*}=\emptyset$, and thus
$\mathcal{Q}\cup\mathcal{Q}^{*}=\mathcal{P}$. We say $\mathcal{Q}$ and
$\mathcal{Q}^{*}$ are a dual pair. Note that $\mathcal{Q}$ completely
determines $\mathcal{Q}^{*}$, and $(\mathcal{Q}^{*})^{*}=\mathcal{Q}$

Figure 26: 1-system of arcs between two punctures

######  Lemma 5.9.

By [1, Theorem 1.7], there are at most three arcs between $\mathcal{Q}$ on
$\Sigma$. Furthermore, if three such arcs $\sigma_{i}$ are contained in a
maximal 1-system $\mathbb{A}$, then they are equivalent to Figure 26, up to a
homeomorphism of $\Sigma$.

###### Proof.

It suffices to prove the second sentence. Up to relabelling, let us assume
$\mathcal{Q}=\\{a,b\\}$. Assume by contradiction all $\sigma_{i}$ are
disjoint. As the three arcs are nonhomotopic,

$\mathcal{Q}\cup\sigma_{1}\cup\sigma_{2}$

bounds two once-punctured bigons, and so it is homotopic to one of
$\sigma_{1},\sigma_{2}$, which is a contradiction. Then, as $\sigma_{3}$ is
disjoint from both $\sigma_{1},\sigma_{2}$, it must lie within one of the two
bigons. But by Lemma 4.2, this is impossible.

Figure 27: $\sigma_{1}$ (blue), $\sigma_{2}$ (orange) with intersection $r$

Let us assume $\sigma_{1},\sigma_{2}$ intersect once, with intersection $r$.
Up to a homeomorphism, the resulting configuration is illustrated in Figure
27, where the colored regions must contain the remaining punctures by the
bigon criterion. Thus, $\sigma_{1},\sigma_{2}$ can be taken to be the orange
and blue arcs on Figure 26.

Suppose by contradiction that $\sigma_{3}$ cannot be homotoped to the green
arc in Figure 26. Thus for some maximal 1-system $\mathbb{A}$ with
$\sigma_{1},\sigma_{2}\in\mathbb{A}$, we have some $\gamma\in\mathbb{A}$ with
$1<|\gamma\cap\sigma_{3}|$. See the first panel of Figure 28 for this
situation.

Figure 28: $\gamma$ in red intersecting $\sigma_{3}$ twice

The top two strands of $\gamma$ must leave the white region. Thus, we have the
second panel of Figure 28. However, at this stage we’ve intersected both
$\sigma_{1}$ and $\sigma_{2}$ once. This implies the bottom strands must stay
in the white region. But this implies $\gamma$ can be homotoped off
$\sigma_{3}$, a contradiction. ∎

######  Lemma 5.10.

For any maximal 1-system $\mathbb{A}$ and dual pair
$\mathcal{Q},\mathcal{Q}^{*}$, there are always exactly four arcs in
$\mathbb{A}$ where each arc is between $\mathcal{Q}$ or $\mathcal{Q}^{*}$.

###### Proof.

We first show for a dual pair, there are no more than $4$ arcs between
$\mathcal{Q}$ or $\mathcal{Q}^{*}$. Assume up to relabelling,
$\mathcal{Q}=\\{a,b\\}$. If there are more than $4$ arcs, necessarily either
$\mathcal{Q}$ or $\mathcal{Q}^{*}$ must contain $3$ arcs by Lemma 5.9. Without
loss of generality, assume $\mathcal{Q}$ contains $3$ arcs. Assume the two
arcs which intersect once are the orange and blue arcs in Figure 26. Thus, we
have more than four arcs iff there exist more than one arc between
$\mathcal{Q}^{*}$. There is one arc, namely the horizontal one, between
$\mathcal{Q}^{*}$ which intersects all three regions, implying it intersects
both colored arcs. Thus, any nonhomotopic arc must necessarily intersect the
orange or blue arc once again. This gives an upper bound of $4$ arcs for each
dual pair. But this in turn gives a lower bound of $4$ arcs, as we must have a
total of $12$ arcs, $3$ choices of $(\mathcal{Q},\mathcal{Q}^{*})$, and no
loops. ∎

######  Corollary 5.11.

For any maximal 1-system $\mathbb{A}$ and dual pair
$\mathcal{Q},\mathcal{Q}^{*}$, exactly one of the two cases occurs:

  1. 1.

There are 3 arcs between $\mathcal{Q}$ and $1$ arc between $\mathcal{Q}^{*}$.

  2. 2.

There are 2 arcs between $\mathcal{Q}$ and $2$ arcs between $\mathcal{Q}^{*}$.

###### Proof.

Combine Lemmas 5.9 and 5.10. ∎

Our next goal is to show that the first case does not extend to a maximal
1-system, while the second case extends to a unique maximal 1-system.

######  Lemma 5.12.

Any 1-system with a $\mathcal{Q}$ such that 3 arcs are between $\mathcal{Q}$
does not extend to a maximal 1-system.

###### Proof.

Up to relabelling, let us assume $\mathcal{Q}=\\{a,b\\}$, and two of the
intersecting arcs to be the orange and blue arcs as in Figure 27. Taking
$\tilde{\mathcal{Q}}=\\{a,d\\}$, thus $\tilde{\mathcal{Q}}^{*}=\\{b,c\\}$, we
see that in neither case can we have more than one arc between
$\tilde{\mathcal{Q}}$ and $\tilde{\mathcal{Q}}^{*}$. The only permissible arc
between $\tilde{\mathcal{Q}}$ and $\tilde{\mathcal{Q}}^{*}$ lies entirely in
the white and orange region respectively. Any other arc would lie in all three
regions, which is disallowed by the pigeonhole principle and the definition of
a 1-system. This contradicts Lemma 5.11. ∎

If fact, our proof shows a bit more. In the remaining case, where both
$\mathcal{Q}$ and $\mathcal{Q}^{*}$ have two arcs between each, the two arcs
between $\mathcal{Q}$ (and similarly $\mathcal{Q}^{*}$) must as in Figure 29.

Figure 29: Two arcs between both $\mathcal{Q},\mathcal{Q}^{*}$

######  Lemma 5.13.

Figure 29 classifies the arcs between two punctures.

###### Proof.

By our above discussion, we know the blue (similarly orange) arcs must be
disjoint. To see the blue arcs determine the orange arcs, it remains to argue
each orange arc must intersect exactly one blue arc. Note that any orange arc
must intersect at least one blue arc, to be between $\mathcal{Q}^{*}$. The
orange arc intersects at most one blue arc, else we contradict the definition
of a 1-system. ∎

Figure 30: $J_{0}$ extended from pyramid

###### Proof of Proposition 5.8.

We first prove that every maximal 1-systems contains the blue, orange and gray
arcs in Figure 31. By Lemma 5.13, we can take the blue and orange arcs to be
in $\mathbb{A}$.

Figure 31: Four regions (white, green, yellow, purple) with gray arcs

The blue and orange arcs divide $\Sigma$ into four regions as in Figure 31.
Note also the four gray arcs which are entirely contained in a region. Let us
show that the gray arc in the yellow region is contained in $\mathbb{A}$; the
same argument will show the other gray arcs are also contained in
$\mathbb{A}$.

Figure 32: Gray arc must be contained in $\mathbb{A}$

Suppose not, so for some maximal 1-system $\mathbb{A}$, there exists some arc
which intersects the gray arc twice. This is illustrated in Figure 32 by the
red arc. The second panel “zooms in” to show which curves the red arc must
intersect by definition of 1-system. The last panel “zooms out” to show the
red arc cannot possibly close up, as it has already intersected with each of
the four arcs once. This yields the desired contradiction.

Figure 33: Arcs in Figure 31

Now that we know that $\mathbb{A}$ contains the arcs in Figure 31, we will
prove that $\mathbb{A}$ uniquely extends to Figure 30. We begin by observing
the arcs in Figure 31 are exactly the yellow arcs forming the pyramid and the
two green arcs in Figure 30. Thus, in the following, we say faces to denote
faces of the pyramid as in 30.

Figure 34: Unique Maximal 1-system $J_{0}$

The arcs in Figure 31 unroll into Figure 33, a parallelogram. We will discuss,
given $\mathcal{Q}$, what can be the arc between $\mathcal{Q}$ in $\mathbb{A}$
distinct from an edge of the pyramid which is guaranteed by Lemma 5.12. An arc
through one face is homotopic to an arc in 31, and through three implies the
arc is a loop. Thus for $\mathcal{Q}=\\{a,d\\}$, it must be one of the two
diagonals of the parallelogram. Tilting the parallelogram to a rectangle, we
see these two choices are equivalent up to a reflection through the vertical
line through $b,c$. Without loss of generality, let us choose the one
indicated by dark orange in Figure 34. But this choice determines an arc
between $b,c$ by Lemma 5.13 as indicated by the light orange arcs. It remains
to choose an arc between $a,c$. Arcs through four faces either intersect the
green or orange arc more than once. Thus, the only arc between $a,c$ is the
one through two faces as seen in the blue arc in Figure 34. This uniquely
determines the arc between $b,d$ indicated by light blue, and we see this is
equivalent to Figure 30. ∎

## References

  * [1] Piotr Przytycki  “Arcs intersecting at most once”  In _Geometric and Functional Analysis_ 25.2, 2015, pp. 658–670
  * [2] Christopher Smith and Piotr Przytycki  “Arcs on spheres intersecting at most twice”  In _Indiana University Math Journal_ 68.1, 2019, pp. 157–178


# The weight distribution of codes over finite chain rings

Giulia Cavicchioni  Department of Mathematics
University of Trento
Italy<EMAIL_ADDRESS>and  Alessio Meneghetti  Department of
Mathematics
University of Trento
Italy<EMAIL_ADDRESS>
###### Abstract.

In this work, we determine new linear equations for the weight distribution of
linear codes over finite chain rings. The identities are determined by
counting the number of some special submatrices of the parity-check matrix of
the code. Thanks to these relations we are able to compute the full weight
distribution of codes with small Singleton defects, such as MDS, MDR and AMDR
codes.

###### Key words and phrases:

Ring-linear code, Weight distribution

###### 2020 Mathematics Subject Classification:

94B05,13M99

##  1\. Introduction

Ring-linear coding theory has been widely studied because of its theoretical
and practical interest. On one hand, ring-linear codes are relevant from an
algebraic perspective: as shown in [5], some optimal but non-linear binary
codes can be represented as linear codes over $\mathbb{Z}/4\mathbb{Z}$ endowed
with the Lee metric. On the other hand, ring-linear codes have received
attention in cryptographic community. The recent effort among cryptographers
to obtain secure post-quantum ciphers [1, 2, 4, 11, 12, 15] led to an increase
in the interest in computationally hard algebraic problems, and an interested
reader can refer to [3, 13, 20] for more details. Code-based cryptography is
one of the most studied and promising areas in the post-quantum framework.
However, due to the necessity of reducing the public key size associated to a
code-based cryptosystem, there has been interest in exploring different
ambient spaces and metrics other then vector spaces over finite fields
equipped with the Hamming metric. For example, codes over finite rings
equipped with the Lee metric may decrease the-public key size of the
cryptosystems; for further details see [7, 21, 22].
Understanding the minimum distance of a code is computationally hard and it is
one of the main problem in Coding Theory. In 1997 Vardy proved that, given a
basis of a code, determining precisely the minimum distance of a linear code
is NP-hard [20]. Hence this computational problem, as well as several related
questions, is linked to the security of post-quantum cryptographic protocols.
Even the problem of calculating the weight distribution of a linear code,
which implies the determination of the minimum distance, is NP-hard. In this
paper we treat the problem of computing the weight distribution of linear
codes over finite chain rings equipped with the Hamming metric. In classical
coding theory the most fundamental result about weight distributions are the
MacWilliams identities, which express how the weight enumerators of a linear
code and its dual relate to each other. Several authors have generalized this
work in different directions. For example, a MacWilliams theorem for codes
over finite Frobenius rings was given by Wood in 1999 [23].
Here, we provide new linear equations for the weight distribution of ring-
linear codes by counting the number of some special submatrices of the parity-
check matrix of the code. This task is certainly as difficult as the original
one; however, it allows to investigate codes having special structure in their
parity-check matrix. The provided equations and MacWilliams identities seems
to be independent, but there could be a possible link in between this
equations and some variant of MacWilliams identities.
This paper is organized as follows. In Section 2 we recall some basics on
linear codes over finite fields. In Section 3 we introduce ring-linear codes;
we investigate the structure of the parity-check matrix and the weight
distribution of a linear code. In Section 4 we derive new relations for the
weight distribution of ring-linear codes; we discuss the optimality of the
result. The obtained formula is a modification of the formula given in [14,
Proposition 5] for linear codes over finite fields and specialised in [17] for
Hermitian codes. In Section 5 we apply our formula to verify the known results
about the distribution of MDS codes. Moreover we derive the weight
distribution formula for MDR and AMDR codes. Finally, in Section 6, we discuss
the connection between MacWilliams identities and the provided relations.

##  2\. Preliminaries on Linear codes over finite fields

In its most general setting, Coding Theory is the study of discrete sets
equipped with a metric. The most studied case is that of algebraic varieties
living in vector spaces over finite fields, and the metric is the Hamming
metric. In this framework, a (linear) code $C$ is a vector subspace of
dimension $k$ of $\left(\mathbb{F}_{q}\right)^{n}$, where the elements of the
code are called codewords and the parameters $n$ and $k$ are respectively
known as the length and the dimension of $C$. The Hamming metric, also known
as Hamming distance, is a discrete metric counting the number of non-zero
coordinates, namely,

$\mathrm{d}(v,w)=\left|\left\\{i\mid v_{i}\neq w_{i},\;1\leq i\leq
n\right\\}\right|\;,$

for any $v=(v_{1},\ldots,v_{n})$ and $w=(w_{1},\ldots,w_{n})$ in
$\left(\mathbb{F}_{q}\right)^{n}$.
Notice that in this work we consider the elements of vector spaces and modules
to be row vectors, a standard notation in Coding Theory. If $v$ is any (row)
vector, then its transpose $v^{\top}$ is a column vector.
The third most important parameter of a code is the so-called minimum distance
$d$, which is the minimum among the Hamming distances of any pair of distinct
codewords, i.e.

$d=\min_{c_{1},c_{2}\in C}\mathrm{d}(c_{1},c_{2})\;,$

and it coincides with the minimum weight of a codeword. The importance of the
minimum distance is related to the capability of codes to correct errors. If
we are presented with a vector $v$ that should be a codeword $c$ of a given
code $C$, even if its coordinates are corrupted (hence $v\notin C$), then we
can safely reconstruct $c$ from $v$ provided that the number of erroneous
coordinates of $v$ is bounded by $\lfloor\frac{d-1}{2}\rfloor$.
The weight distribution of a code of length $n$ specifies the number of
codewords of each possible weight $0,1,\dots,n$. Even if the weight
distribution does not in general uniquely determine a code, it gives important
information: in addition to providing the correction capability of a code, it
allows to calculate the probability of undetected errors (see [19, Chapter
2]).

##  3\. Generalities on linear codes over finite chain rings

A finite ring with unity $1\neq 0$ is called a left (resp. right) chain ring
if its left (resp. right) ideals are linearly ordered by inclusion. Note that
a finite chain ring is a local ring where all the ideals are principal.
Throughout the paper let $R$ be a finite commutative chain ring. Let $\gamma$
be the generator of the maximal ideal and let $s$ be its the nilpotency index.
Let $\mathbb{K}$ denote the residue field with $p$ elements $R/\gamma R$.

######  Definition 3.1.

A _linear code of length $n$ in the alphabet $R$_ is a submodule of $R^{n}$.
The free module $R^{n}$ is called the _ambient space_ of the code.

######  Definition 3.2.

The _Hamming weight_ of an element $c=(c_{1},\dots,c_{n})\in R^{n}$ is the
number $w(c)$ of non-zero entries of $c$.

######  Definition 3.3.

A matrix $G$ is called a _generator matrix_ for the code $C$ over $R$ if the
rows of $G$ span $C$ and none of them can be written as a linear combination
of the other rows of $G$.

As shown in [16], any linear code over a finite chain ring has a generator
matrix. In our framework it is convenient to work with a generator matrix in
standard form.

######  Proposition 3.4.

[16, Proposition 3.2] Let $C$ be a linear code in $R^{n}$. $C$ is permutation
equivalent to a code having the following generator matrix in standard form:

$G=\begin{bmatrix}I_{k_{0}}&A_{0,1}&A_{0,2}&A_{0,3}&\dots&A_{0,s-1}&A_{0,s}\\\
0&\gamma I_{k_{1}}&\gamma A_{1,2}&\gamma A_{1,3}&\dots&\gamma A_{1,s-1}&\gamma
A_{1,s}\\\
0&0&\gamma^{2}I_{k_{2}}&\gamma^{2}A_{2,3}&\dots&\gamma^{2}A_{2,s-1}&\gamma^{2}A_{2,s}\\\
\vdots&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&0&\dots&\gamma^{s-1}I_{k_{s-1}}&\gamma^{s-1}A_{s-1,s}\\\ \end{bmatrix}\
,$

where $A_{i,s}\in M_{k_{i}\times n-K}(R/\gamma^{s-i}R)$ and $A_{i,j}\in
M_{k_{i}\times k_{j}}(R/\gamma^{s-i}R)$ for $j<s$.

For all $0\leq i\leq s$ the $k_{i}$’s denote the number of rows of $G$ that
are divisible by $\gamma^{i}$ but not by $\gamma^{i+1}$. The parameters
$k_{0},\dots,k_{s-1}$ are the same for all generator matrices in systematic
form, and $C$ is said to be of _type_ $(k_{0},k_{1},\dots,k_{s-1})$. The
_rank_ of $C$ is defined as $K=\sum_{i=0}^{s-1}k_{i}$.

######  Definition 3.5.

The _free rank_ of $C$ is defined to be the maximum of the rank of the free
submodules of $C$ and it coincides with $k_{0}$.

######  Definition 3.6.

A linear code is said to be _free_ if its rank coincides with its free rank.
In this case, the code is a free $R$-submodule which is isomorphic to
$R^{k_{0}}$.

If $C$ is a free code, then any systematic generator matrix has the form

$G=\begin{bmatrix}I_{k_{0}}&A\end{bmatrix}\in(\mathbb{Z}/p^{s}\mathbb{Z})^{k_{0}\times
n}.$

Since for all $0\leq j\leq s-1$ we have $\lvert\gamma^{j}R\rvert=p^{s-j}$ (see
[16, Lemma 2.4]), it is possible to compute the cardinality of a linear code.

######  Theorem 3.7.

[16, Theorem 3.5] A linear code $C$ over $R$ of type $(k_{0},\dots,k_{s-1})$
has cardinality $\lvert C\rvert=p^{\sum_{i=0}^{s-1}(s-i)k_{i}}$.

We attach the standard inner product to the ambient space i.e.
$v\cdot w=\sum v_{i}w_{i}$. The dual code $C^{\perp}$ of $C$ is defined, as in
the classical case, by

$C^{\perp}=\\{v\in R^{n}\mid v\cdot w=0\text{ for all }w\in C\\}.$

In [23], Wood proved that the dual code of a code over a Frobenius ring, and
hence over a finite chain ring, is well defined (i.e. $(C^{\perp})^{\perp}=C)$
. The dual code $C^{\perp}$ has the following parameters:

######  Theorem 3.8.

Let $C\subseteq R^{n}$ be a linear code of rank $K$ and type
$(k_{0},\dots,k_{s-1})$. Then $C^{\perp}$ is a linear code of rank $n-k_{0}$
and type $(n-K,k_{s-1},\dots,k_{1})$.

As a consequence, the dual code of a free code is again free.

We call any matrix $H$ a _parity-check_ matrix for $C$ if its kernel is $C$.

######  Proposition 3.9.

[16, Theorem 3.10] Let $C$ be a linear code of type $(k_{0},\dots,k_{s-1})$.
Then $C$ is permutation equivalent to a code having a parity-check matrix in
systematic form:

(3.1) $H=\begin{bmatrix}B_{0,s}&B_{0,s-1}&\dots&B_{0,1}&I_{n-K}\\\ \gamma
B_{1,s}&\gamma B_{1,s-1}&\dots&\gamma I_{k_{s-1}}&0\\\
\vdots&\vdots&&\vdots&\vdots\\\
\gamma^{s-1}B_{s-1,s}&\gamma^{s-1}I_{k_{1}}&\dots&0&0\\\
\end{bmatrix}=\begin{bmatrix}H^{(0)}\\\ \gamma H^{(1)}\\\ \vdots\\\
\gamma^{s-1}H^{(s-1)}\par\par\par\end{bmatrix},$

where, for $0\leq i,j\leq s,\
B_{i,j}=-\sum_{k=i+1}^{j-1}B_{i,k}A^{T}_{s-j,s-k}-A^{T}_{s-j,s-i}\ .$

Clearly, $H$ is a generator matrix for $C^{\perp}$.

###  3.1. Hamming weight distribution and Singleton-like bounds

As in the classical case, the _Hamming-weight distribution_ of a ring-linear
code is a vector $A=(A_{i})_{i=0,1\dots,n}$, where $A_{i}$ denotes the number
of codewords of $C$ of weight $i$. The weight distribution can be encoded as
coefficients in a polynomial.

######  Definition 3.10.

The (Hamming)-_weight enumerator polynomial_ of a ring-linear code $C$ of
length $n$ is the bivariate polynomial

$W_{C}(X,Y)=\sum_{c\in C}X^{n-w(c)}Y^{w(c)}=\sum_{i=0}^{n}A_{i}X^{n-i}Y^{i}\
.$

The Hamming-weight enumerators of a code and its dual are related by the
MacWilliams identities.

######  Theorem 3.11.

[23, Theorem 8.3] For linear codes over a finite chain ring $R$ with $p^{s}$
elements, the MacWilliams identities hold:

$W_{C^{\perp}}(X,Y)=\frac{1}{\lvert C\rvert}W_{C}(X+(p^{s}-1)Y,X-Y)\ .$

Moreover, for the Hamming metric over $R$, Singleton-like bounds are known.

######  Remark 3.12.

The Singleton bound for codes over any alphabet of size $p^{s}$ states that

$d\leq n-\log_{p^{s}}(\lvert C\rvert)+1\ ,$

(see for example [10]). In the framework of codes over finite chain rings,
only free codes meet this bound and they are said _maximum distance separable_
(MDS) codes.
As shown in [6], for codes over principal ideal rings

(3.2) $d\leq n-K+1\ .$

This bound is in general tighter than the Singleton bound and they coincides
if and only if the code is free. A linear code over a finite chain ring
meeting this bound is said to be _maximum distance with respect to rank_
(MDR). In particular a code $C$ is MDS if and only if it is MDR and free.

It is well-known ([18, Corollary 1]) that the MDS property is invariant under
duality (i.e. the dual of an MDS code over a finite chain ring is again MDS).
In general, the dual code of an MDR code does not preserve the property.

######  Example 3.13.

Let
$C=\langle(1,0,1),(0,2,0),(0,0,2)\rangle\subset(\mathbb{Z}/4\mathbb{Z})^{3}$
be a linear code. $C$ is MDR since $d(C)=1=n-K+1$. However, its dual code
$C^{\perp}=\langle(2,0,2),(0,2,0)\rangle$ has minimum distance
$d(C^{\perp})=1<2=(n-K+1)$.

As in the classical case of linear codes over finite fields, we can measure
how far away a linear code $C$ is from being MDR.

######  Definition 3.14.

Let $C$ be a linear code of length $n$ and rank $K$. The _defect_ $s(C)$ of
$C$ is defined as $s(C)\coloneqq n+1-K-d$.

###  3.2. On the parity-check matrix and its submatrices

The parity-check matrix of a code give important information on some
structural properties of the code, such as the minimum distance.
Analogously to linear codes over finite fields (see [8, Theorem 1.4.13] ),
given a code $C$ over a finite chain ring there is a link between the weights
of the codewords of $C$ and its parity-check matrix $H$.

######  Theorem 3.15.

Let $C$ be a linear code over $R$ with parity-check matrix $H$. If $c\in C$,
the columns of $H$ corresponding to the non-zero coordinates of $c$ are
linearly dependent. Conversely, if a linear dependence relation with only non-
zero coefficients exists among $w$ columns of $H$, then there is a codeword in
$C$ of weight $w$ whose non-zero coordinates correspond to these columns.

###### Proof.

If $c\in C$, the matrix product $0=Hc^{T}=\sum_{i=1}^{n}\mathbf{h_{i}}c_{i}$,
where $\mathbf{h_{i}}$ is the $i^{th}$ column of $H$, is a linear combination
of the columns of $H$ with coefficients provided by $c$. Conversely, if there
are $w$ linearly dependent columns in $H$, then
$\sum_{i=0}^{n}\alpha_{i}\mathbf{h_{i}}=0,\ \alpha_{i}\in R$ and $w$ of them
are non-zero . If $c=(\alpha_{1},\dots,\alpha_{n})$, then $Hc^{\top}=0$ and
$\text{w}(c)=w$ and $c$ is the desired codeword. ∎

For any $0\leq j\leq s-1$, let $H^{(j)}$ be submatrices of $H$ defined
according to (3.1). Since the $H^{(j)}$s are all full rank, the following is
immediate.

######  Proposition 3.16.

Let $H$ be a parity-check matrix for the code $C$, and hence a generator
matrix for $C^{\perp}$. A message $v\in R^{n-k_{0}}$ is encoded as the zero
codeword in $C^{\perp}$ if and only if it is of the form

$v=[\underbrace{0,\dots,0}_{n-K}||\gamma^{s-1}\cdot\underbrace{v_{1}}_{k_{s-1}}||\gamma^{s-2}\cdot\underbrace{v_{2}}_{k_{s-2}}||\dots||\gamma\cdot\underbrace{v_{s}}_{k_{1}}]\
.$

######  Notation 3.17.

Let $M\in M_{t\times n}(R)$ be a matrix. According to the notation of
Proposition 3.4, we say that $M$ is of _type_ $(t_{0},\dots,t_{s-1})$ if
$t_{i}$ rows of $M$ are divisible by $\gamma^{i}$ but not by $\gamma^{i+1}$,
$0\leq i\leq s-1$.

######  Definition 3.18.

Let $M\in M_{t\times n}(R)$ be a matrix of type $(t_{0},\dots,t_{s-1})$.

  * •

For any subset $\mathcal{I}\subseteq\\{1,\dots,n\\}$ of size $\nu$,
$\mathcal{I}=\\{i_{1},\dots i_{\nu}\\}$ with
$i_{1}<i_{2}<\dots<i_{\nu}$, we define $M_{[\mathcal{I}]}$ as the $t\times\nu$
submatrix of $M$ identified by the columns indices $\mathcal{I}.$

  * •

We define $N_{M}(\nu,r_{0},r_{1},\dots,r_{t})$ to be the number of
$t\times\nu$ submatrices of $M$ of type $(r_{0},r_{1},\dots,r_{t})$

Let $C$ be an $R$-linear code of type $(k_{0},k_{1},\dots,k_{s-1})$, and let
$H\in M_{(n-k_{0})\times n}(R)$ be a parity-check matrix in standard form for
$C$. For any fixed $\mathcal{I}$ of size $\nu$, $H_{[\mathcal{I}]}$ is a
$(n-k_{0})\times\nu$ submatrix of $H$ and, as in (3.1), we can write:

(3.3) $H_{[\mathcal{I}]}=\begin{bmatrix}H_{[\mathcal{I}]}^{(0)}\\\ \gamma
H_{[\mathcal{I}]}^{(1)}\\\ \vdots\\\
\gamma^{s-1}H_{[\mathcal{I}]}^{(s-1)}\par\par\par\end{bmatrix}\ ,$

where each $H_{[\mathcal{I}]}^{(j)}$ is obtained from $H^{(j)}$ by removing
the unnecessary columns. Since we are selecting $\nu<n$ columns from $H$, the
type of $H_{[\mathcal{I}]}$ and $H$ may differ. First of all, note that the
dimension of the first block may decrease. Indeed:

  1. (1)

Some rows of $H_{[\mathcal{I}]}^{(0)}$ can be written as linear combinations
of the others, so they can be removed from the parity-check matrix;

  2. (2)

Some rows of $H_{[\mathcal{I}]}^{(0)}$ are multiples of $\gamma^{l}$ for some
$1\leq l\leq s-1$. If this is the case, the rows can be moved in one of the
subsequent blocks.

On the other hand, the dimension of the second block can either increase,
decrease or remain unchanged. One or more of the following may occur:

  1. (1)

Some rows of $H_{[\mathcal{I}]}^{(0)}$ are added to $H_{[\mathcal{I}]}^{(1)}$;

  2. (2)

Some rows of of $H_{[\mathcal{I}]}^{(1)}$ are multiples of $\gamma^{l},\ 2\leq
l\leq s-1$. In this case, collecting $\gamma^{l}$, the row can be moved in one
of the subsequent blocks;

  3. (3)

Some rows of $H_{[\mathcal{I}]}^{(1)}$ are linear combinations of the others,
and thus they can be removed from the matrix.

The procedure can be iterated for any other block $H_{[\mathcal{I}]}^{(j)}$,
$2\leq j\leq s-1$. Therefore, the type of $H_{[\mathcal{I}]}$, being different
from the type of $H$, can not be studied in its full generality. However, if
$\nu$ satisfies certain conditions, the structure of $H_{[\mathcal{I}]}$
become more clear: we will focus on this problem in the next section.

##  4\. Weight distribution for linear codes

From now on, let $C$ be a linear code over a finite chain ring $R$ of length
$n$, rank $K$ and type $(k_{0},k_{1},\dots,k_{s-1})$.

######  Lemma 4.1.

Let $H\in M_{(n-k_{0})\times n}(R)$ be a parity-check matrix for the code $C$.
If $n-d^{\perp}<\nu\leq n$, then all the $(n-k_{0})\times\nu$ submatrices of
$H$ are of type $(n-K,k_{s-1},k_{s-2},\dots,k_{1})$. In particular they all
have rank $n-K$.

###### Proof.

Being the parity-check matrix of a linear code of type
$(k_{0},k_{1},\dots,k_{s-1})$, $H$ is of type
$(n-K,k_{s-1},k_{s-2},\dots,k_{1})$. Let $H_{[\mathcal{I}]}$,
$\mathcal{I}=\\{i_{1},\dots,i_{\nu}\\}$, be a $(n-k_{0})\times\nu$ submatrix
of $H$. Without loss of generality $H_{[\mathcal{I}]}$ can be written as in
(3.3). By contradiction, assume that a row in $H^{(0)}_{[\mathcal{I}]}$ is
multiple of $\gamma$. Without loss of generality we may assume it is the first
one. If $\bar{v}=[\gamma,0,\dots,0]\in R^{n-K}$, then $\bar{v}\cdot
H^{(1)}_{[\mathcal{I}]}=0$. The vector $\bar{v}$ can be used as a first brick
for constructing a new vector $v$. By Proposition 3.16, $v$ is not encoded as
the zero codeword. Therefore $c\coloneqq v\cdot H$ is a codeword in
$C^{\perp}$ of weight $w(c)\leq n-\nu$, contradicting the hypothesis
$d^{\perp}>n-\nu$. So, since no row in $H^{(0)}_{[\mathcal{I}]}$ is a multiple
of $\gamma$, in $H_{[\mathcal{I}]}$ there are at most $k_{s-1}$ rows which are
multiple of $\gamma$ but not of $\gamma^{2}$. Applying to
$H^{(1)}_{[\mathcal{I}]}$ the procedure described above, it is possible to
show they are exactly $k_{s-1}$. Iterating the process for all the remaining
blocks $H^{(2)}_{[\mathcal{I}]},\dots,H^{(s-1)}_{[\mathcal{I}]}$ we get the
thesis. ∎

######  Corollary 4.2.

Let $C$ be a linear code with parity-check matrix $H$ and $n-d^{\perp}<\nu\leq
n.$

$N_{H}(\nu,t_{0},t_{1},\dots t_{s-1})=\begin{cases}\binom{n}{\nu}\text{\quad
if }t_{0}=n-K\text{ and }t_{i}=k_{s-i},\ 1\leq i\leq s-1\\\ \hskip
5.0pt0\quad\text{\hskip 5.0ptotherwise}\end{cases}.$

######  Proposition 4.3.

Let $C$ be a code of type $(k_{0},\dots,k_{s-1})$ with parity-check matrix
$H\in M_{n-k_{0}\times n}(R)$. Let $\\{{A_{i}}\\}_{i=0,\dots,n}$ be the weight
distribution of $C$. If $n-d^{\perp}<\nu\leq n$, then

(4.1) $\sum_{l=0}^{\nu}\binom{n-l}{\nu-l}A_{l}=\binom{n}{\nu}\frac{\lvert
C\rvert}{p^{s(n-\nu)}}\ .$

###### Proof.

Let $V_{[\mathcal{I}]}$ be the kernel of $H_{[\mathcal{I}]}$. Consider the map

$\varphi_{[\mathcal{I}]}\colon V_{[\mathcal{I}]}\rightarrow R^{n},\
\quad\varphi_{[\mathcal{I}]}(v)=(\bar{v}_{1},\dots,\bar{v}_{n}),\quad\bar{v}_{j}=\begin{cases}v_{j}\text{
if }j\in\\{1,\dots,\nu\\}\\\ 0\hskip 5.0pt\text{ otherwise}\end{cases}.$

$\varphi_{[\mathcal{I}]}$ is the restriction of the map

$\varphi_{\nu}\colon\bigsqcup_{\mathcal{I}\colon\lvert\mathcal{I}\rvert=\nu}V_{[\mathcal{I}]}\to
C$

to $V_{[\mathcal{I}]}$. The elements in the domain of $\varphi_{\nu}$ can be
counted in two different ways:

  1. (1)

For any choice of $\mathcal{I}=\\{i_{1},\dots,i_{\nu}\\}$, by Lemma 4.1, the
matrix $H_{[\mathcal{I}]}$ is of length $\nu$ and of type
$(n-K,k_{s-1},k_{s-2},\dots,k_{1})$. Therefore $H_{[\mathcal{I}]}$ is a
parity-check matrix of a linear code $C^{\prime}$ of type
$(\nu-n+k_{0},k_{1},\dots,k_{s-1})$ and length $\nu$. Thus, by Theorem 3.7,
$C^{\prime}$ has with $p^{s(\nu-n+k_{0})+\sum_{i=1}^{s-1}(s-i)k_{i}}$
elements. Hence, by Corollary 4.2,

(4.2)
$\begin{split}\left|\bigsqcup_{\mathcal{I}\colon\lvert\mathcal{I}\rvert=\nu}V_{[\mathcal{I}]}\right|&=N_{H}(\nu,n-K,k_{s-1},k_{s-2},\dots,k_{1})\lvert
V_{[\mathcal{I}]}\rvert=\\\
&=\binom{n}{\nu}p^{s(\nu-n+k_{0})+\sum_{i=1}^{s-1}(s-i)k_{i}}\ .\end{split}$

  2. (2)

We consider a codeword $c\in C$ of weight $l\leq\nu.$ Let
$\mathcal{I}_{1}=\text{supp}(c)$. Any choice of $\nu-l$ indices
$\mathcal{I}_{2}\subset\\{1,\dots,n\\}\smallsetminus\mathcal{I}_{1}$
identifies uniquely an element in
$\sqcup_{\mathcal{I}\colon\lvert\mathcal{I}\rvert=\nu}V_{[\mathcal{I}]}.$ More
precisely, $\mathcal{I}_{1}\cup\mathcal{I}_{2}$ determines uniquely
$H_{[\mathcal{I}_{1}\cup\mathcal{I}_{2}]}$, clearly
$c_{[\mathcal{I}_{1}\cup\mathcal{I}_{2}]}\in
V_{[\mathcal{I}_{1}\cup\mathcal{I}_{2}]},$ and so there is an unique element
$v\in V_{[\mathcal{I}_{1}\cup\mathcal{I}_{2}]}$ such that
$\varphi_{[\mathcal{I}_{1}\cup\mathcal{I}_{2}]}(v)=c$, that is
$v=c_{[\mathcal{I}_{1}\cup\mathcal{I}_{2}]}$. In order to determine the size
of $\varphi^{-1}_{\nu}(c)$, the fiber of $c$ under the map $\varphi_{\nu}$, it
is enough to count all possible subsets of
$\\{1,\dots,n\\}\smallsetminus\mathcal{I}_{1}$ with size $\nu-l$. It follows
that the fiber of each codeword of weight $l$ has $\binom{n-l}{\nu-l}$
elements, and we observe that all the fibers of such codewords form a
partition of
$\sqcup_{\mathcal{I}\colon\lvert\mathcal{I}\rvert=\nu}V_{[\mathcal{I}]}$.
Since there are $A_{l}$ codeword of weight $l$, we obtain

(4.3)
$\left|\bigsqcup_{\mathcal{I}\colon\lvert\mathcal{I}\rvert=\nu}V_{[\mathcal{I}]}\right|=\sum_{l=0}^{\nu}\binom{n-l}{\nu-l}A_{l}\
.$

Putting together (4.2) and (4.3) we get (4.1). ∎

######  Corollary 4.4.

For a free code $C$ of length $n$ and rank $k_{0}=K$ over $R$ the weight
distribution formula reads:

$\sum_{l=0}^{\nu}\binom{n-l}{\nu-l}A_{l}=\binom{n}{\nu}p^{s(\nu+K-n)}\ .$

######  Theorem 4.5.

Let $\sigma$ be the sum of the Singleton defects of $C$ and $C^{\perp}$. the
knowledge of $\sigma+d+K-k_{0}-1$ elements of the weight distribution
$\\{A_{0},\dots,A_{n}\\}$ is enough to compute the full weight distribution of
$C$ and $C^{\perp}$. In particular the knowledge of $d$ and of any
$\sigma+K-k_{0}-1$ elements of $\\{A_{d},\dots,A_{n}\\}$ is enough to compute
the entire weight distribution of $C$ and $C^{\perp}$.

###### Proof.

Consider equation (4.1) with $\nu$ varying in range
$\\{n-d^{\perp}+1,\dots,n\\}$. We obtain a linear system of the form

$\mathcal{P}\cdot A(C)=b\ ,$

where $\mathcal{P}$ is a truncated Pascal matrix with $d^{\perp}$ rows and $n$
columns. By [9], all the minors of $\mathcal{P}$ of order $d^{\perp}$ are non-
zero. Hence, the knowledge of $n-d^{\perp}+1=K-\sigma-1+d-(K-k_{0})$ elements
in the weight distribution allow us to obtain a linear system that admit a
unique solution. Finally, the knowledge of $d$ implies the knowledge of
$A_{0},\dots,A_{d-1}$. Hence, it is sufficient to know other
$n-d^{\perp}+1-d=\sigma+K-k_{0}-1$ elements in $\\{A_{d},\dots A_{n}\\}$ to
determine the full weight distribution of the code. ∎

We now show that, at least in some cases, Theorem 4.5 is optimal. Indeed, in
general, it is not possible to deterministically deduce the weight
distribution of a linear code with less then $\sigma+K-k_{0}-1$ elements in
$\\{A_{0},\dots,A_{n}\\}$.

######  Definition 4.6.

A linear code is said to be _Almost-MDR code_(AMDR) if it has Singleton defect
equal to 1. An AMDR code whose dual is still AMDR is called _Near-MDR_.

######  Definition 4.7.

A free code is said to be _Almost-MDS code_(AMDS) if it has Singleton defect
equal to 1. An AMDS code whose dual is still AMDS is called _Near-MDS_

Consider two Near-MDS codes with the same parameters. For a Near-MDS only
$\sigma+K-k_{0}-1=1$ weight is necessary to determine the full weight
distribution. If the sole knowledge of $d$ and $n$ was enough to compute the
entire weight distribution of the code, then any two near-MDS would be
formally equivalent. But this is in general false:

######  Example 4.8.

Let $C_{1}$ and $C_{2}$ be two near-MDS codes over $\mathbb{Z}_{5^{3}}$ of
length $n=4$, rank $K=2$ and minimum Hamming distance $d=2$, generated
respectively by

$G_{1}=\begin{bmatrix}1&0&57&0\\\ 0&1&0&68\\\ \end{bmatrix},$

and

$G_{2}=\begin{bmatrix}1&0&5&43\\\ 0&1&82&5\\\ \end{bmatrix}.$

Their weight distributions are respectively:

$\mathcal{A}_{1}=(1,0,248,0,15376)\ ,$

and

$\mathcal{A}_{2}=(1,0,8,480,15136)\ .$

Therefore $C_{1}$ and $C_{2}$ are not formally equivalent.

##  5\. Weight distribution of codes with small Singleton defects

Theorem 4.5 lead us to further investigate codes having a small number of
Singleton defects.

We start focusing on codes meeting the Singleton bound: from Remark 3.12
follows that $\sigma=0$ for any MDS code.
The weight distribution of MDS codes is well known (see [18, Theorem 5]);
however it can be directly obtained with the sole knowledge of the length and
the minimum distance from Theorem 4.5:

######  Theorem 5.1.

Let $C$ be a MDS code of length $n$ and rank $k$, then

$A_{w}(C)=\binom{n}{w}\sum_{j=0}^{w-d}(-1)^{j}\binom{i}{j}(p^{s-d+1-j}-1)\ .$

Now we move to linear codes meeting the generalized Singleton bound (3.2), the
MDR codes. As shown in 3.12 the dual of an MDR code is not necessarily MDR.
Therefore the weight distribution may depend on one or more parameters.
Let $C$ be an MDR code of length $n$, rank $K$, minimum distance $d=n-K+1$ and
let $C^{\perp}$ be its dual having rank $n-K$ and minimum distance
$k_{0}-\sigma+1$ for some $\sigma\geq 0$. According to the notation of
Proposition 4.1, since $\nu<n-d^{\perp}$ and $d^{\perp}=k_{0}-\sigma+1$, we
can write $\nu=n-k_{0}+\sigma+i$ with $i$ ranging in
$\\{0,\dots,k_{0}-\sigma\\}.$ Let $q=\frac{\lvert C\rvert}{p^{s(n-\nu)}}$.
Then equation (4.1) becomes

$\sum_{l=0}^{n-k_{0}+\sigma-1+i}\binom{n-l}{n-k_{0}+\sigma+i-l}A_{l}=\binom{n}{n-k_{0}+\sigma+i}q\
.$

Since $A_{0}=1$ and $A_{l}=0$ for all $1\leq l\leq n-k$, we may write

$\displaystyle\binom{n}{n-k_{0}+\sigma+i}+\sum_{h=0}^{\sigma-k+k_{0}-2}\binom{k-1-h}{\sigma+i+K-k_{0}-h}A_{n+K+1+h}+$
$\displaystyle+\sum_{l=n+\sigma-
k_{0}}^{n-k_{0}+\sigma+i}\binom{n-l}{n-k_{0}+\sigma+i-l}A_{l}=\binom{n}{n-k_{0}+\sigma+i}q\
.$

Let $l=n+\sigma-k_{0}+j$, then

$\displaystyle\sum_{j=0}^{i}\binom{k_{0}-\sigma-j}{i-j}A_{n+\sigma-k_{0}+j}=$
$\displaystyle=\binom{n}{n-k_{0}+\sigma+i}$
$\displaystyle(q-1)-\sum_{h=0}^{\sigma+K-k_{0}-2}\binom{K-1-h}{K-k_{0}+\sigma+i+h}A_{n+j+1+h}\
.$

We can re-write the linear system in matrix form, as $\mathcal{P}\cdot A=b$
with $\mathcal{P}$ Pascal matrix
$\Big{[}\binom{k_{0}-\sigma-j}{i-j}\Big{]}_{i,j=0,\dots,k_{0}-\sigma}$.
Therefore
$\mathcal{P}^{-1}=(-1)^{i-j}\Big{[}\binom{k_{0}-\sigma-j}{i-j}\Big{]}_{i,k=0,\dots,k_{0}-\sigma}$.
More explicitly:

######  Proposition 5.2.

Let $C$ be an MDR code of length $n$, rank $K$, minimum distance $d=n-K+1$ and
let $C^{\perp}$ be its dual having rank $n-k_{0}$ and minimum distance
$k_{0}-\sigma+1$ for some $\sigma\geq 0$. Let $q=\frac{\lvert
C\rvert}{p^{s(n-\nu)}}$. The knowledge of
$\\{A_{n-K+1},\dots,A_{n-k_{0}+\sigma-1}\\}$ in the weight distribution of $C$
is enough to compute the entire weight distribution of $C$.
In particular, for all $0\leq i\leq k_{0}-\sigma-K$, we have:

$\displaystyle A_{n-k_{0}+\sigma+i}=\sum_{j=0}^{i}(-1)^{i-j}$
$\displaystyle\binom{k_{0}-\sigma-j}{i-j}\Bigg{[}\binom{n}{n-k_{0}+\sigma+i}(q-1)-$
$\displaystyle-\sum_{h=0}^{\sigma+K-k_{0}-2}\binom{k-h-1}{k-k_{0}+\sigma+i-h-1}A_{n+k+1+h}\Bigg{]}\
.$

In a similar fashion we can derive the weight distribution of an AMDR code:

######  Proposition 5.3.

Let $C$ be an AMDR code of length $n$, rank $K$, minimum distance $d=n-K$ and
let $C^{\perp}$ be its dual having rank $n-k_{0}$ and minimum distance
$k_{0}-\sigma+1$ for some $\sigma\geq 0$. Let $q=\frac{\lvert
C\rvert}{p^{s(n-\nu)}}$. The knowledge of
$\\{A_{n-K+1},\dots,A_{n-k_{0}+\sigma-1}\\}$ is enough to compute the entire
weight distribution of $C$.
In particular, for all $0\leq i\leq k_{0}-\sigma-K$ we have:

$\displaystyle A_{n-k_{0}+\sigma+i}=\sum_{j=0}^{i}(-1)^{i-j}$
$\displaystyle\binom{k_{0}-\sigma-j+1}{i-j}\Bigg{[}\binom{n}{n-k_{0}+\sigma+i-1}(q-1)-$
$\displaystyle\sum_{h=0}^{\sigma+K-k_{0}-2}\binom{k-h}{k-k_{0}+\sigma+i-h+1}A_{n+k+h}\Bigg{]}\
.$

Clearly, by specializing the previous formula, we also get the weight
distributions of Near-MDS and Near-MDR codes.

##  6\. Relation with MacWilliams identities

Both in classical and ring-linear coding theory, the most fundamental result
about weight distributions are the MacWilliams identities (Theorem 3.11). They
relate the weight enumerator polynomial of a linear code and its dual.
However, in our framework it is more convenient to work with other equivalent
set of equations in place of the polynomial form of 3.11. Following the
outline of [10, Chapter 5, Section 2], and combining it with 3.11 we can
deduce the following equality:

$\sum_{j=0}^{n-\nu}\binom{n-j}{\nu}A_{j}=\frac{\lvert
C\rvert}{p^{s\nu}}\sum_{j=0}^{\nu}\binom{n-j}{n-\nu}A^{\perp}_{j}\ ,\ \text{
for }\ 0\leq\nu\leq n.$

Moreover, in a similar fashion to [8, Theorem 7.2.3], we get:

(6.1) $\sum_{j=0}^{n}\binom{j}{\nu}A_{j}=\frac{\lvert
C\rvert}{p^{s\nu}}\sum_{j=0}^{\nu}(-1)^{j}\binom{n-j}{n-\nu}(p^{s}-1)^{\nu-j}A_{j}^{\perp}\
,\ \text{ for }\ 0\leq\nu\leq n.$

If $\nu<d^{\perp}$, each $A_{j}^{\perp}$ of the right hand side of (6.1) is
equal to zero except for $A_{0}^{\perp}$ which is equal to 1. Therefore we get
a ring-variant of Pless’ equations.

######  Proposition 6.1.

For any $\nu<d^{\perp}$

(6.2) $\sum_{j=0}^{n}\binom{j}{\nu}A_{j}=\frac{\lvert
C\rvert}{p^{s\nu}}\binom{n}{n-\nu}(p^{s}-1)^{\nu},\ \text{ for }\
0\leq\nu<d^{\perp}.$

Hence, when enough terms of the weight distributions are known, systems in 4.3
and 6.1 are equivalent.

######  Corollary 6.2.

Let $\sigma$ be the sum of the Singleton defects of $C$ and $C^{\perp}$. Using
equation (6.2), the knowledge of $\sigma+d+K-k_{0}-1$ elements of the weight
distribution $\\{A_{0},\dots,A_{n}\\}$ is enough to compute the full weight
distribution of $C$ and $C^{\perp}$. In particular the knowledge of $d$ and of
any $\sigma+K-k_{0}-1$ elements of $\\{A_{d},\dots,A_{n}\\}$ is enough to
compute the entire weight distribution of $C$ and $C^{\perp}$.

###### Proof.

The proof follows the same outline of Proposition 4.5. ∎

Therefore, the two systems of equations (4.3) and (6.2) are equivalent
provided the existence of the code.

##  7\. Conclusion

In analogy to linear codes over finite fields, the minors of the parity-check
matrix of a ring-linear code enable us to determine linear relations between
the weights of the codes. Our formulae enable to verify the weight
distribution of MDS codes. Moreover this result allows to determine the full
weight distributions of MDR, Near-MDR, AMDR codes.
The number of parameters necessary to derive the full weight distribution of a
code and its dual depends on the the sum of the Singleton defects of the code
and its dual, an in particular it is bounded by $n+K-k_{0}-1$.
An interesting extension of this work would be the study of more classes of
codes, either by considering the case of non-AMDR codes or families obtained
via structured parity check matrices. A second promising line of research
would be the derivation of formulas for weight distribution related to
different metrics, e.g. Lee metric or Rank metric.

## Acknowledgement

The publication was created with the co-financing of the European Union - FSE-
REACT-EU, PON Research and Innovation 2014-2020 DM1062 / 2021. The authors are
member of the INdAM Research Group GNSAGA. The core of this work was partially
presented on a talk given at COMBINATORICS 2022 in Mantova, Italy by the first
author.

## References

  * [1] Nicolas Aragon, Paulo SLM Barreto, Slim Bettaieb, Loic Bidoux, Olivier Blazy, Jean-Christophe Deneuville, Philippe Gaborit, Shay Gueron, Tim Guneysu, Carlos Aguilar Melchor, et al.  Bike: bit flipping key encapsulation.  2017\.
  * [2] Roberto Avanzi, Joppe Bos, Léo Ducas, Eike Kiltz, Tancrède Lepoint, Vadim Lyubashevsky, John M Schanck, Peter Schwabe, Gregor Seiler, and Damien Stehlé.  Crystals-kyber algorithm specifications and supporting documentation.  NIST PQC Round, 2(4):1–43, 2017.
  * [3] Elwyn Berlekamp, Robert McEliece, and Henk Van Tilborg.  On the inherent intractability of certain coding problems (corresp.).  IEEE Transactions on Information Theory, 24(3):384–386, 1978.
  * [4] Daniel J Bernstein.  Introduction to post-quantum cryptography.  In Post-quantum cryptography, pages 1–14. Springer, 2009.
  * [5] AR Calderbank, AR Hammons Jr, P Vijay Kumar, NJA Sloane, and P Solé.  The z4-linearity of kerdock, preparata, goethals and related codes.  IEEE Trans. Inf. Theory, 40(2):301–319, 1994.
  * [6] Steven T Dougherty and Keisuke Shiromoto.  Mdr codes over $\mathbb{Z}_{k}$.  IEEE Transactions on Information Theory, 46(1):265–269, 2000.
  * [7] Anna-Lena Horlemann-Trautmann and Violetta Weger.  Information set decoding in the lee metric with applications to cryptography.  arXiv preprint arXiv:1903.07692, 2019.
  * [8] W Cary Huffman and Vera Pless.  Fundamentals of error-correcting codes.  Cambridge university press, 2010.
  * [9] Scott N Kersey.  Invertibility of submatrices of pascal’s matrix and birkhoff interpolation.  arXiv preprint arXiv:1303.6159, 2013.
  * [10] Florence Jessie MacWilliams and Neil James Alexander Sloane.  The theory of error correcting codes, volume 16.  Elsevier, 1977.
  * [11] Robert J McEliece.  A public-key cryptosystem based on algebraic.  Coding Thv, 4244:114–116, 1978.
  * [12] Carlos Aguilar Melchor, Nicolas Aragon, Slim Bettaieb, Loıc Bidoux, Olivier Blazy, Jean-Christophe Deneuville, Philippe Gaborit, Edoardo Persichetti, Gilles Zémor, and IC Bourges.  Hamming quasi-cyclic (hqc).  NIST PQC Round, 2(4):13, 2018.
  * [13] Alessio Meneghetti, Alex Pellegrini, and Massimiliano Sala.  On the equivalence of two post-quantum cryptographic families.  Annali di Matematica Pura ed Applicata (1923-), pages 1–25, 2022\.
  * [14] Alessio Meneghetti, Marco Pellegrini, and Massimiliano Sala.  A formula on the weight distribution of linear codes with applications to amds codes.  Finite Fields and Their Applications, 77:101933, 2022.
  * [15] Dustin Moody, Gorjan Alagic, Daniel C Apon, David A Cooper, Quynh H Dang, John M Kelsey, Yi-Kai Liu, Carl A Miller, Rene C Peralta, Ray A Perlner, et al.  Status report on the second round of the nist post-quantum cryptography standardization process.  2020\.
  * [16] Graham H Norton and Ana Sălăgean.  On the structure of linear and cyclic codes over a finite chain ring.  Applicable algebra in engineering, communication and computing, 10(6):489–506, 2000.
  * [17] Marco Pellegrini and Massimiliano Sala.  Weight distribution of hermitian codes and matrices rank.  Finite Fields and Their Applications, 60:101578, 2019.
  * [18] Keisuke SHIROMOTO.  Note on mds codes over the integers modulo $p^{m}$.  Hokkaido Mathematical Journal, 29(1):149–157, 2000.
  * [19] Klove Torleiv.  Codes for error detection, serial on coding theory and cryptography, vol. 2, 2007.
  * [20] Alexander Vardy.  The intractability of computing the minimum distance of a code.  IEEE Transactions on Information Theory, 43(6):1757–1766, 1997\.
  * [21] Violetta Weger, Massimo Battaglioni, Paolo Santini, Franco Chiaraluce, Marco Baldi, and Edoardo Persichetti.  Information set decoding of lee-metric codes over finite rings.  arXiv preprint arXiv:2001.08425, 2020.
  * [22] Violetta Weger, Karan Khathuria, Anna-Lena Horlemann, Massimo Battaglioni, Paolo Santini, and Edoardo Persichetti.  On the hardness of the lee syndrome decoding problem.  arXiv preprint arXiv:2002.12785, 2020.
  * [23] Jay A Wood.  Duality for modules over finite rings and applications to coding theory.  American journal of Mathematics, pages 555–575, 1999.


LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Univ.
Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon,
France Department of Physics, Imperial College London, SW7 2AZ London, UK
School of Physics and Astronomy, Queen Mary University of London, London E1
4NS, UK

# The rotation angle distribution underlying magnetic field fluctuations in
the 1/f range of solar wind turbulent spectra

L. Matteiniins:xins:y  D. Stansbyins:y  T.S. Horburyins:y C.H.K.
Chenins:zins:y ins:xins:xins:yins:yins:zins:z

###### Abstract

We discuss properties of large amplitude magnetic field fluctuations during
fast Alfvénic solar wind streams, focussing on the statistics of the rotation
angle between consecutive magnetic field vector measurements for different
scales in the plasma. Since in the fast solar wind fluctuations preserve the
modulus of the magnetic field to a good approximation, the tip of the magnetic
field vector is observed to move on a sphere of approximately constant radius
$|\bf{B}|$. We then compare statistics of solar wind measurements with that of
a simple model of a random walk bounded on a spherical surface. The analogy
consists in the fact that in both systems the geometrical constraint imposes a
limiting amplitude at large separations and thus introduces a break scale in
the power spectrum of the fluctuations, leading to a shallower slope for
scales where the fluctuations amplitude becomes scale-independent. However,
while in the case of the random walk the saturation of the fluctuations occurs
when the pattern becomes uniform on the sphere (flat distribution of the
cosine of the rotation angle), transitioning then to a white noise regime, in
the solar wind magnetic field fluctuations saturate in amplitude maintaining a
preferential direction. We suggest that this behaviour is due to the presence
of the background interplanetary magnetic field, which keeps some long-range
memory in the system also when the fluctuations becomes independent of the
scale. This long-range correlation is a necessary ingredient in order to
produce the 1/f spectrum observed at large scales in the solar wind.

##  1 Introduction

In a recent paper [1] we have discussed properties of the large scale solar
wind magnetic spectrum, focussing on the so called 1/f range. This is the
portion of the power spectrum of the fluctuations which corresponds to scales
above the turbulent Kolmogorov inertial range and during fast streams it is
characterised by a $-1$ spectral slope [2, 3]. The origin of this range and
its spectral shape are still debated [4, 5, 6, 7]. We briefly summarise here
the main idea behind our model: in magnetised plasmas, if imposing a regime of
low magnetic compressibility (as typically observed in the fast solar wind) we
expect that there is a limiting amplitude for the fluctuations $\delta\bf{B}$
when they reach order of the background field $\bf{B}$; indeed, if
$|\delta\bf{B}|\gg|B|$ fluctuations in the components induce also changes in
the magnetic field intensity, so that they become highly compressible. Note
that in the fast solar wind, at large scales $|\delta{\bf B}|/|{\bf B}|\sim 1$
but $\delta|\bf{B}|\ll|\delta\bf{B}|$, so that the fluctuations are nearly
incompressible ($|\bf{B}|$ is conserved). In [1] we suggest that in a
turbulent plasma, when the amplitude of the fluctuations becomes large so that
at some scale $l_{0}$ $|\delta{\bf B}|\sim|\bf{B}|$, a saturation occurs to
prevent the their further growth at scales $l>l_{0}$, and thus maintaining the
plasma at a low level of magnetic compressibility. This leads to a break in
the spectrum at $l_{0}$ and a constant level $|\delta\bf{B}|/|\bf{B}|\sim 1$
for scales $l>l_{0}$. Remarkably, this simple phenomenological scenario
applies well to the large amplitude fluctuations of the solar wind; moreover
in situ spacecraft measurements show that the scale $l_{0}$ at which
$|\delta\bf{B}|/|\bf{B}|$ reaches order unity in the solar wind plasma always
identifies the spectral break between 1/f and inertial ranges as observed at
various radial distances from the Sun [1].

An important comment is in order here. The existence of a flat first order
structure function over some scales (constant $|\delta\bf{B}|$) does not
directly imply a unique spectral slope. For example, if the fluctuations of
this range are completely uncorrelated (white noise), the associated power
spectrum is also flat. On the other hand, when the autocorrelation function of
the fluctuations is not identically zero (some long range correlation exists
in the system), then the usual connection between second order structure
functions at scale $l$, $\delta B_{l}^{2}$, and power spectral density (PSD)
for k-vectors $k=1/l$, $P(k)$, can be made:

$\delta B_{l}^{2}=P(k)\cdot k$ (1)

so that if the former has slope $\alpha$, the latter has slope $\alpha-1$ [8].
It is then straightforward to see that our model, based on the saturation of
the structure function $\delta B_{l}$ above a scale $l_{0}$ ($\alpha=0$),
directly predicts a $-1$ spectrum. Also note that for eq. 1, $-1$ is the
steepest possible slope.

However, as mentioned, this is possible only if some long-range correlations
are present in the plasma at large scales [9], otherwise a flat (white noise)
spectrum would be expected. The aim of this paper is then to investigate
further this aspect in solar wind observations, and compare the result with a
simple model of a bounded random walk.

##  2 A bounded random walk model

The fact that a self-similar process, which is in principle scale invariant,
may be characterised by a transition between different regimes at a particular
scale, should not be surprising. Consider in fact random walks on a 2-D
surface; the two-point correlation for a random walk ensemble leads to a power
law distribution with slope $-2$. This is however true if the walk takes place
on an unbounded surface, so that separations at arbitrary large scales can be
performed. The large scale correlation in this case preserves the same
properties as the small scale ones, leading to a single power-law spectrum. On
the other hand, consider now the case when the random walk develops on a
spherical surface. While on distances (separation times) that are small
compared to the curvature radius of the surface 2-point correlations still
display the same $-2$ spectrum as in the unbounded case, large scale
correlations are characterised by a different behaviour. Indeed, if the random
walk is let evolve for sufficiently long time, the pattern starts to cover
uniformly the sphere, and the trajectory will eventually come back to close to
the starting point.

An example of this dynamics is shown in fig. 1 where we report results from a
simple model [10] which tracks the motion of a vector whose tip is constrained
on a sphere and subject to a random walk. The model is let evolve for $2^{17}$
time steps of $\Delta\theta=0.1$rad, and results are taken as averages of 10
ensembles. The left panel shows the PDFs of the rotation angle $\theta$
between two distinct vectors for increasing time lags (i.e. number of steps
$N$), from red to dark blue; at sufficiently large time lags, the distribution
of $\cos{(\theta)}$ becomes flat, meaning a uniform coverage of the sphere, as
expected, and does not evolve further when increasing the time lag. This
suggests that, although the underlying process generating the pattern is self-
similar, the statistics made on time separations that are larger than the
typical time needed to walk around the full sphere is different from that of
smaller scale. Large scales are influenced by the fact that the walk occurs on
a closed surface, and once this uniformly covered, they follow a different
law.

This behaviour obviously affects the spectral properties of the system. In the
right panel of fig. 1 we show the power spectrum of the Cartesian components
of the resulting separation time series. We can see that at short time scales
a spectral slope of index $-2$ is recovered, as expected for a pure random
walk; on the contrary, at a scale of $N\sim 10^{3}$ time steps $t_{0}$,
corresponding approximately to the time needed to cover the whole sphere
($N\sim 4\pi/\Delta\theta^{2}$) and indicated by the vertical solid line,
there is a change in the slope leading to a flat spectrum, as expected for
vanishing correlation between points uniformly distributed (i.e. white noise).

Figure 1: Results from a random walk model bounded on a spherical surface.
Left: PDF of the cosine of the rotation angle $\theta$ between 2 vectors at
different time separations encoded by colours, lag increasing from red to
blue. Right: Power spectrum of the resulting time series of the random walk
displacements on the sphere (top) and associated local spectral slopes
(bottom). The expected break scale, corresponding to the number of steps
needed to cover uniformly the sphere, is indicated by the solid line the in
top panel.

The one above is just a simple example of how the presence of a geometrical
constraint can lead to a break in the self-similarity and thus introduce a
change in the expected power law of the fluctuations. Although it does not
apply directly to turbulent interplanetary magnetic field fluctuations, which
cannot be modelled as a simple random walk, this is still an instructive
example to compare with the solar wind observations described in the next
section.

##  3 Solar wind observations

Let us now consider in situ observations from the solar wind. Following [1] we
use Ulysses measurements at high latitudes, when the spacecraft was
continuously embedded in fast polar wind (i.e. very regular, highly Alfvénic
and with little variations of the magnetic and plasma pressures).

Figure 2: PDF of the rotation angle $\theta$ between 2 magnetic field vector
measurements separated by a variable time lag $\Delta t$ in the fast solar
wind. Colours encode different scales, from small (red) to large (blue)
separations. Left and right panels show the distribution of $\theta$ and
normalised $\theta/\left<\theta\right>$, respectively. As a reference,
red/green PDFs belong to the turbulence inertial range (${1<\Delta
t(\rm{s})<10^{3}}$) while almost overlapping blue distributions correspond to
the 1/f range (${\Delta t>5\cdot 10^{3}}$s). The PDF approximatively
corresponding to the break scale separating the two ranges is highlighted with
black diamonds

We focus on the change of orientation between two different magnetic field
measurements $B(t)$ and $B(t+\Delta t)$ as a function of the scale $\Delta t$.
We recall that due to the high speed of the flow, in the solar wind time
measurements (frequencies) really correspond to Doppler-shifted spatial scales
(k-vectors). Following [11] we calculate the PDFs of the angle $\theta$
between the two magnetic field vectors for different time lags $\Delta t$ (see
also [1] for details). The left panel of fig. 2 shows the distribution of
magnetic rotations in the fast solar wind corresponding to Ulysses
measurements at approximately 2 AU, from day 100 to 250 of year 1995. Colours
encode different scales, from small (red/orange) to large (blue/purple)
scales; the PDF approximatively corresponding to the break between inertial
and 1/f ranges ($\Delta t\sim 5\cdot 10^{3}$s) is highlighted with black
diamonds. Due to the small power in the fluctuations at small scales the PDF
of $\theta$ for small time lags peaks at small angles. Rotations become
significantly larger in the inertial range of the turbulence (green curves),
starting to reach the extreme boundary of 180 degrees. At larger scales,
corresponding to the 1/f range (blue/purple), the PDF approaches a quasi-
symmetric shape between 0 and 180 degrees; moreover, PDFs do not evolve
further, as expected for scale-independent fluctuations (consistent with that,
the corresponding PDFs of the fluctuations $\delta B$ also do not change
inside the 1/f range and saturate to a constant level $|\delta{\bf B}|/|{\bf
B}|\sim 1$, see [1]).

The right panel of fig. 2 shows the same distributions each normalised to its
mean value $\left<\theta\right>$. As discussed in [11], at small scales the
normalised PDFs can be reasonably well described by a lognormal distribution;
at these scales the PDFs of $\theta$ (left panel) are sufficiently far from
the 180 degrees boundary, so that a high-tail can be sustained. However,
moving to larger scales and approaching the 1/f range, the effect of the
boundary on the lognormal tail is apparent. As a consequence, the distribution
of $\theta/\left<\theta\right>$ in the 1/f has a more Gaussian-like shape,
although some skewness is visible.

In order to better appreciate the spread of rotation angles, the left panel of
fig. 3 shows the distribution of $\cos{\theta}$ for all scales, with the same
colour code as fig. 2. This goes from a strongly peaked shape at small scales
-where angles are almost always smaller than a few degrees- to shallower
distributions at larger scales. Once more, note that curves corresponding to
the 1/f range all show the same shape and fall on top of each other. However,
unlike fig. 1, the PDFs never reach a flat shape, indicative of a uniform
distribution over the sphere of constant $|B|$; indeed, a clear asymmetry
between positive and negative values of $\cos{\theta}$ is maintained at all
scales, including the 1/f range. As a consequence, rotations are
preferentially distributed around the direction of the background field
(approximately the Parker spiral). This implies that even at the largest
scales (in the 1/f range, where $|\delta{\bf B}|/|{\bf B}|\sim 1$) the
fluctuations are not totally uncorrelated and have some long-range
correlation, the memory of the background field direction.

In [1] we have shown that the saturated shape observed for the 1/f range,
corresponds to an approximatively exponential relation:

$PDF(\cos{\theta})\sim\exp{(\alpha\cos{\theta})}$ (2)

where $\alpha$ is an empirical constant, whose value can be obtained from the
observations. Relation (2) is shown in fig. 3 as a dashed red line; it seems
to describe reasonably well the main part of the PDF in the 1/f range. For the
Ulysses observations, we get $\alpha\sim 0.8$. As a comparison, we show in the
right panel of the same figure the same analysis for a fast solar wind stream
observed by the Helios spacecraft at 0.3 AU (see [1] for more details). Also
in this case we recover the same qualitative picture as for Ulysses (taken at
$\sim 2$AU); for Helios however, the PDFs in the 1/f are less shallow, and
$\alpha\sim 1.8$. This suggests a possible evolution of the PDF saturated
shape with increasing radial distance.

Figure 3: Left: PDF of the cosine of the rotation angle $\theta$ between 2
magnetic field vectors in the fast solar wind at 2 AU (Ulysses); colors encode
different scales as in previous figure. At the largest scales (1/f range,
blue) the distribution tends to become shallower, with a roughly constant
slope, although not entirely flat; this implies that the distribution of the
underlying magnetic field vector tends to spread over the full sphere of
radius $|\bf{B}|$, but not yet uniformly. Right: same analysis for the fast
wind at 0.3 AU (Helios measurements).

##  4 Discussion and conclusion

To summarise, we have discussed properties of large scales fluctuations in the
solar wind, when $|\delta{\bf B}|/|{\bf B}|\sim 1$ and the tip of magnetic
field vector approximatively moves on a sphere of constant radius $|{\bf B}|$
[12, 13], and compared them to a simple model based on a random walk bounded
on a sphere. In particular we have focussed on the rotation angle $\theta$
between two different vectors separated by a variable time lag, to probe the
effect of the geometrical boundary on the statistics of different scales.

We have found that the difference in the two cases lies in the condition that
characterises and constrains the asymptotic state. In the simple random walk
model, the self-similarity is broken by a purely geometrical condition, i.e.
the pattern of the walk becomes uniform over the sphere. The appearance of a
spectral break is then caused by the lack of two-point correlations above a
certain scale (the size of the sphere) and as a consequence the associated
spectrum becomes flat, i.e. white noise.

On the other hand, in the solar wind case some physical mechanism must be
responsible for the saturation of the amplitude of the fluctuations typically
observed at large scales [14]; in [1] we propose that this is related to a
constraint of low magnetic compressibility in the plasma. As for the bounded
random walk, the saturation of the amplitude of the fluctuations naturally
introduces a break in the solar wind magnetic spectrum. However, unlike fig.
1, the distribution of $\cos{\theta}$ in the fast solar wind never reaches a
flat shape (fig. 3), suggesting that a preferential direction common to all
fluctuations continue to exist: this is the interplanetary background magnetic
field. This maintains some level of long-range correlation between the
fluctuations, as required to build a 1/f spectrum like that observed in the
solar wind at large scales.

###### Acknowledgements.

This work was supported by national program PNST of CNRS/INSU co-funded by
CNES. DS and TH are supported by STFC grants ST/K001051/1 and ST/N000692/1,
respectively. CHKC is supported by STFC Ernest Rutherford Fellowship
ST/N003748/2.

## References

  * [1] Matteini L., Stansby D., Horbury T. S. Chen C. H. K., ApJ Lett.2018.
  * [2] Bavassano B., Dobrowolny M., Mariani F. Ness N. F., J. Geophys. Res.8719823617.
  * [3] Denskat K. U. Neubauer F. M., J. Geophys. Res.8719822215.
  * [4] Matthaeus W. H. Goldstein M. L., Physical Review Letters571986495.
  * [5] Velli M., Grappin R. Mangeney A., Phys. Rev. Lett.6319891807.
  * [6] Verdini A., Grappin R., Pinto R. Velli M., ApJ Lett.7502012L33.
  * [7] Chandran B. D. G., Journal of Plasma Physics842018905840106.
  * [8] Monin A. S. Yaglom A. M., Statistical fluid mechanics: Mechanics of turbulence. Volume 2, Cambridge, Mass., MIT Press 1975.
  * [9] Keshner M. S., IEEE Proceedings701982212.
  * [10] Barnes A., J. Geophys. Res.8619817498.
  * [11] Chen C. H. K., Matteini L., Burgess D. Horbury T. S., MNRAS4532015
  * [12] Bruno R., Carbone V., Primavera L., Malara F., Sorriso-Valvo L., Bavassano B. Veltri P., Annales Geophysicae2220043751.
  * [13] Matteini L., Horbury T. S., Pantellini F., Velli M. Schwartz S. J., ApJ802201511.
  * [14] Mariani F., Ness N. F., Burlaga L. F., Bavassano B. Villante U., J. Geophys. Res.8319785161.