# Quasi-Perfect and Distance-Optimal Codes in the Sum-Rank Metric

Hao Chen Hao Chen is with the College of Information Science and
Technology/Cyber Security, Jinan University, Guangzhou, Guangdong Province,
510632, China (e-mail: haochen@jnu.edu.cn). The research of Hao Chen was
supported by NSFC Grant 62032009.

###### Abstract

Constructions of quasi-perfect and distance-optimal codes are interesting and
challenging problems. In this paper, we give the following three results.

1) If $\lambda|q^{sm}-1$ and
$\lambda<\sqrt{\frac{(q^{s}-1)}{2(q-1)^{2}(1+\epsilon)}}$, an infinite family
of distance-optimal $q$-ary sum-rank codes with the block length
$t=\frac{q^{sm}-1}{\lambda}$, the matrix size $s\times s$, the cardinality
$q^{s^{2}t-s(2m+3)}$ and the minimum sum-rank distance four is constructed.

2) Block length $q^{4}-1$ and the matrix size $2\times 2$ distance-optimal
sum-rank codes with the minimum sum-rank distance four and the Singleton
defect four are constructed. These sum-rank codes are close to the sphere
packing bound , the Singleton-like bound and have much larger block length
$q^{4}-1>>q-1$.

3) For given positive integer $m$ satisfying $2\leq m$, an infinite family of
quasi-perfect sum-rank codes with the matrix size $2\times m$, and the minimum
sum-rank distance three is also constructed. Families of quasi-perfect binary
sum-rank codes with the matrix size $2\times 2$ and the minimum sum-rank
distance four are also presented.

We also show that new distance-optimal binary sum-rank codes can be obtained
from the Plotkin sums.

Index terms: Quasi-perfect sum-rank code. Distance-optimal sum-rank code.
Singleton defect. Plotkin sum of sum-rank codes.

##  1 Introduction

###  1.1 Preliminaries

Sum-rank codes have found wide applications in multishot network coding, see
[27, 33, 34], space-time coding, see [36], and coding for distributed storage,
see [25, 24]. There have attracted many attentions in recent years, see
e.g.[6, 24, 26, 27, 28, 31, 32] and references therein. Fundamental properties
and some bounds on sizes of sum-rank codes were given in the paper [6]. For a
nice survey of sum-rank codes and their applications, we refer to [29].

Let ${\bf F}_{q}^{(n,m)}$ be the set of all $n\times m$ matrices. Let
$n_{i}\leq m_{i}$ be $2t$ positive integers satisfying $m_{1}\geq
m_{2}\cdots\geq m_{t}$. Set $N=n_{1}+\cdots+n_{t}$. Let

${\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}={\bf F}_{q}^{n_{1}\times
m_{1}}\bigoplus\cdots\bigoplus{\bf F}_{q}^{n_{t}\times m_{t}}$

be the set of all ${\bf x}=({\bf x}_{1},\ldots,{\bf x}_{t})$, ${\bf
x}_{i}\in{\bf F}_{q}^{n_{i}\times m_{i}}$, $i=1,\ldots,t$. This is a linear
space over ${\bf F}_{q}$ of the dimension $\Sigma_{i=1}^{t}n_{i}m_{i}$. We
call $n_{1}\times m_{1},\ldots,n_{t}\times m_{t}$, matrix sizes of sum-rank
codes. Set $wt_{sr}({\bf x}_{1},\ldots,{\bf x}_{t})=rank({\bf
x}_{1})+\cdots+rank({\bf x}_{t})$ and

$d_{sr}({\bf x},{\bf y})=wt_{sr}({\bf x}-{\bf y}),$

for ${\bf x},{\bf y}\in{\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$. This
is indeed a metric on ${\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$. This
sum-rank metric satisfies

$d_{sr}({\bf x}+{\bf y},{\bf z}+{\bf y})=d_{sr}({\bf x},{\bf y}).$

Then numbers of elements in balls with a given radius in the sum-rank metric
space is independent of their centers. Let $V_{sr}(q,r)$ be the number of
elements in a ball with the radius $r$, see [6, 29].

A $q$-ary sum-rank code ${\bf C}$ of block length $t$ and matrix sizes
$n_{1}\times m_{1},\ldots,n_{t}\times m_{t}$ is a subset of the finite metric
space ${\bf C}\subset{\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$. Its
minimum sum-rank distance is defined by

$d_{sr}({\bf C})=\min_{{\bf x}\neq{\bf y},{\bf x},{\bf y}\in{\bf
C}}d_{sr}({\bf x},{\bf y}).$

When ${\bf C}\subset{\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$ is a
linear subspace over ${\bf F}_{q}$, ${\bf C}$ is called a linear sum-rank
code. When $q=2$, this code is called a binary sum-rank code. When
$n_{i}=m_{i}=1$, $i=1,\ldots,t$, error-correcting codes in the Hamming metric
are just sum-rank codes with the matrix size $1\times 1$.

The Hamming weight $wt({\bf a})$ of a vector ${\bf
a}=(a_{0},\ldots,a_{t-1})\in{\bf F}_{q}^{t}$ is the cardinality of its
support,

$supp({\bf a})=\\{i:a_{i}\neq 0\\}.$

The Hamming distance $d({\bf a},{\bf b})$ between two vectors ${\bf a}$ and
${\bf b}$ is $d({\bf a},{\bf b})=wt({\bf a}-{\bf b})$. Then ${\bf F}_{q}^{t}$
is a finite Hamming metric space. The minimum (Hamming) distance of a code
${\bf C}\subset{\bf F}_{q}^{t}$ is,

$d=\min_{{\bf a}\neq{\bf b}}\\{d({\bf a},{\bf b}),{\bf a}\in{\bf C},{\bf
b}\in{\bf C}\\}.$

The Hamming metric is just the sum-rank metric on the space
${\mathbf{{F}}}_{q}^{(1,1)}\oplus\cdots\oplus{\mathbf{{F}}}_{q}^{(1,1)}={\mathbf{{F}}}_{q}^{t}$.
One of the main goal of theory of error-correcting codes is to construct codes
${\bf C}\subset{\bf F}_{q}^{t}$ with large cardinalities and minimum
distances. In general, there are some upper bounds on cardinalities or minimum
distances of codes. Optimal codes attaining these bounds are particularly
interesting, see [23].

A linear $q$-ary code with the length $t$, the dimension $k$ (and the minimum
distance $d$), is denoted by a linear $[t,k]_{q}$ (or $[t,k,d]_{q}$) code. A
$q$-ary code with the length $t$, the cardinality $M$, and the minimum
distance $d$, is denoted by an $(t,M,d)_{q}$ code. The Singleton bound for
general codes asserts $M\leq q^{t-d+1}$ and codes attaining this bound is
called (nonlinear) maximal distance separable (MDS) codes. Reed-Solomon codes
are well-known linear MDS codes, see [23]. Construction of non-Reed-Solomon
type linear MDS codes have attracted many attentions, see, e.g., [8] and
references therein. Cyclic and Bose-Chaudhuri-Hocquenghem (BCH) codes are one
of the most important class of codes in theory and practice, see [23, 12].

For a code ${\bf C}$ in the Hamming metric space ${\bf F}_{q}^{t}$, we define
its covering radius by

$R_{covering}({\bf C})=\max_{{\bf x}\in{\bf F}_{q}^{n}}\min_{{\bf c}\in{\bf
C}}\\{wt({\bf x}-{\bf c})\\}.$

Hence the Hamming balls $B(x,R_{covering}({\bf C}))$ centered at all codewords
$x\in{\bf C}$, with the radius $R_{covering}({\bf C})$ cover the whole space
${\bf F}_{q}^{t}$. Then ${\bf C}$ is called a covering code with the radius
$R_{covering}({\bf C})$. We refer to the excellent book [10] on this classical
topic in coding theory.

A code satisfying

$R_{covering}({\bf C})=\lfloor\frac{d({\bf C})-1}{2}\rfloor$

is perfect, see [37, 23, 10]. It was proved that perfect codes have the same
parameters as Hamming codes and Golay codes, see [37, 10]. We refer to [15]
for nonlinear perfect codes. The next best possibilities are quasi-perfect
codes. A code ${\bf C}$ satisfying

$R_{covering}({\bf C})=\lfloor\frac{d({\bf C})-1}{2}\rfloor+1$

is called quasi-perfect, see [23, Section 5, Chapter 1] and [38, 14, 15].
Quasi-perfect codes are obviously ideal candidates for these parameters with
which there is no perfect code.

The sphere packing bound for $(t,M,d)_{q}$ codes asserts that

$M\cdot V_{H}(q,\lfloor\frac{d-1}{2}\rfloor)\leq q^{t},$

where $V_{H}(q,r)=1+t(q-1)+\displaystyle{t\choose
2}(q-1)^{2}+\cdots+\displaystyle{t\choose r}(q-1)^{r}$ is the volume of the
ball with the radius $r$ in the Hamming metric space ${\bf F}_{q}^{t}$, since
balls centered at codewords with the radius $\lfloor\frac{d-1}{2}\rfloor$ are
disjoint, see [23]. If there is an $(t,M,d)_{q}$ code, and there is no
$(t,M,d+1)_{q}$ code, this $(t,M,d)_{q}$ code is called distance-optimal. Some
distance-optimal codes are with respect to the sphere packing bound. That
means, there is an $(t,M,d)_{q}$ code and $M\cdot
V_{H}(q,\lfloor\frac{d}{2}\rfloor)>q^{t}$. For a linear $[t,k,d]_{q}$ code
with the codimension $r=t-k$ satisfying

$V_{H}(q,\frac{d}{2})>q^{r},$

this linear $[t,k,d]_{q}$ code is distance-optimal with respect to the sphere
packing bound.

The Singleton-like bound for sum-rank codes with the matrix size $n\times m$,
$n\leq m$ is

$|{\bf C}|\leq q^{m(nt-d_{sr}+1)}.$

Similar to the case in Hamming metric, the difference $m(nt-d_{sr}({\bf
C})+1)-\log_{q}|{\bf C}|$ is called the Singleton defect of the sum-rank code
${\bf C}$. A sum-rank code attaining this bound (Singleton defect zero) is
called a maximum sum-rank-metric distance (MSRD) code. When $t\leq q-1$ and
$N\leq(q-1)m$, MSRD codes attaining the above Singleton-like bound were
constructed in [24]. They are called linearized Reed-Solomon codes, we also
refer to [31] for the further results. We also refer to [6, Section VII] and
[1] for other results on MSRD codes. Notice that Singleton defects are
$0,s,2s,3s,\ldots$. Singleton defect $s$ sum-rank codes with the matrix size
$s\times s$ can be called almost MSRD codes. MSRD codes over finite chain
rings was constructed in a recent paper [30].

The sphere packing bound for a sum-rank code ${\bf C}$ with $M$ codewords and
the minimum sum-rank distance $d_{sr}$ asserts

$M\cdot V_{sr}(q,\lfloor\frac{d_{sr}-1}{2}\rfloor)\leq
q^{n_{1}m_{1}+\cdots+n_{t}m_{t}},$

see [6]. A sum-rank code in ${\bf F}_{q}^{n_{1}\times
m_{1}}\bigoplus\cdots\bigoplus{\bf F}_{q}^{n_{t}\times m_{t}}$, with the
minimum sum-rank distance $d_{sr}$ and $M$ codewords is called distance-
optimal, if there is no sum-rank code with the minimum sum-rank distance
$d_{sr}+1$ and $M$ codewords. It is obvious if there is a sum-rank code ${\bf
C}$ with the minimum sum-rank distance $d_{sr}$ and $M$ codewords, and

$V_{sr}(q,\lfloor\frac{d_{sr}}{2}\rfloor)\cdot
M>q^{n_{1}m_{1}+\cdots+n_{t}m_{t}},$

this code ${\bf C}$ is distance-optimal with respect to the sphere packing
bound. Therefore a linear (over ${\bf F}_{q}$) sum-rank code with the
codimension $r$ is distance-optimal with respect to the sphere packing bound,
if

$V_{sr}(q,\lfloor\frac{d_{sr}}{2}\rfloor)>q^{r},$

is satisfied. A sum-rank code ${\bf C}\subset{\bf
F}_{q}^{(n_{1},m_{1})}\oplus\cdots\oplus{\bf F}_{q}^{(n_{t},m_{t})}$ with the
minimum sum-rank distance $d_{sr}$ satisfying

$V_{sr}(q,\lfloor\frac{d_{sr}-1}{2}\rfloor)\cdot|{\bf
C}|=q^{n_{1}m_{1}+\cdots+n_{t}m_{t}}$

is a perfect sum-rank code. The whole space is the disjoint union of balls
with the radius $\lfloor\frac{d_{sr}-1}{2}\rfloor$ centered at all codewords
of a perfect code ${\bf C}$.

Let ${\bf C}\subset{\bf F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$ be a code
in the sum-rank metric, its covering radius $R_{sr}({\bf C})$ is the smallest
radius $R_{sr}$ such that the balls

$B_{sr}({\bf x},R_{sr})=\\{{\bf y}\in{\bf
F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}:wt_{sr}({\bf y}-{\bf x})\leq
R_{sr}\\},$

centered at all codewords ${\bf x}\in{\bf C}$, cover the whole space ${\bf
F}_{q}^{(n_{1},m_{1}),\ldots,(n_{t},m_{t})}$. Perfect sum-rank codes are just
these codes satisfying

$R_{sr}({\bf C})=\lfloor\frac{d_{sr}({\mathbf{{C}}})-1}{2}\rfloor.$

As in the case of the sum-rank metric with the matrix size $1\times 1$ (the
Hamming metric), a sum-rank code satisfying

$R_{sr}({\bf C})=\lfloor\frac{d_{sr}({\mathbf{{C}}})-1}{2}\rfloor+1$

is called quasi-perfect. Quasi-perfect sum-rank codes are naturally extensions
of quasi-perfect codes of the matrix size $1\times 1$ in the Hamming metric.
Then how to construct quasi-perfect sum-rank codes with the matrix size
$n\times m$, where both $n$ and $m$ are larger than $1$, is an interesting and
challenging problem.

###  1.2 Related works and our contributions

Some distance-optimal ternary cyclic codes with the length $3^{m}-1$ and
minimum distance four were constructed in [42, 11]. Distance-optimal codes
with minimum distance four and six were constructed in [42, 11, 17, 18, 41].
There are only few distance-optimal codes with minimum distance six reported
in the literature, see [17] and [39, Theorem 9]. Several infinite families of
optimal codes with respect to the Griesmer bound were constructed in [20]. For
distance-optimal sum-rank codes, we refer to [26, 9]. Distance-optimal
(actually perfect) sum-rank codes with minimum sum-rank distance three were
constructed in [26]. Perfect sum-rank codes constructed by U. Martínez-Peñas,
in [26] have the matrix size $1\times n$. He asked in [26, Remark 5] if there
is perfect sum-rank codes for the matrix size $m\times n$, $m>1,n>1$. On the
other hand, the first infinite family of binary cyclic sum-rank codes with
minimum sum-rank distance four was constructed in [9, Section 7].

The constructions of quasi-perfect codes for various parameters has received
many attentions. Many binary, ternary linear or nonlinear quasi-perfect codes
and quasi-perfect linear codes over large fields were constructed and
classified, see [13, 14, 15, 22, 2, 38, 16, 22]. In particular, short quasi-
perfect codes over ${\bf F}_{q}$ with the minimum distance four were
constructed in [16].

In this paper, we construct infinitely many new families of distance-optimal
$q$-ary cyclic codes with the minimum distance four. These families of
distance-optimal sum-rank codes have new parameters, compared with previous
distance-optimal codes constructions. Infinitely many families of distance-
optimal $q$-ary cyclic sum-rank codes with minimum distance four are
presented. An infinite family of distance-optimal $q$-ary sum-rank codes with
the block length $q^{4}-1$, the matrix size $2\times 2$ and the minimum sum-
rank distance four is constructed. The Singleton defect of these distance-
optimal sum-rank codes is $4$. However block lengths $q^{4}-1$ is much larger
than $q-1$, when compared with MSRD codes in [24, 31] with the zero Singleton
defect and the block length $q-1$. It is showed that Plotkin sums of sum-rank
codes lead to more distance-optimal sum-rank codes.

More importantly, an infinite family of quasi-perfect $q$-ary sum-rank codes
with the matrix size $2\times m$, where $m$ is a positive integer satisfying
$2\leq m$, and the minimum sum-rank distance three, is constructed. We also
construct the matrix size $2\times 2$ quasi-perfect binary sum-rank codes with
the minimum sum-rank distance four from quasi-perfect codes over
${\mathbf{{F}}}_{4}$ given in [16]. These are the first several families of
quasi-perfect sum-rank codes with the matrix size $n\times m$, where both $m$
and $n$ are larger than $1$.

The paper is organized as follows. In Section 2, we recall basic facts about
cyclic codes, which are used for constructions in this paper. A basic
construction proposed in our previous paper [7] was also included. In Section
3, distance-optimal $q$-ary cyclic codes with the minimum distance four are
constructed. In Section 4 and 5, distance-optimal cyclic sum-rank codes with
the minimum sum-rank distance four are constructed. In Section 6, quasi-
perfect sum-rank codes with the matrix size $2\times m$ are constructed. In
Section 8, quasi-perfect binary sum-rank codes with the minimum sum-rank
distance four are constructed. In Section 8, Plotkin sums of sum-rank codes
are introduced and new distance-optimal binary sum-rank codes are constructed.
Section 9 concludes the paper.

##  2 Cyclic codes and the construction of sum-rank codes from codes in the
Hamming metric

Cyclic codes was introduced by E. Prange in 1957, see [35]. They are one of
the most important class of linear codes in coding theory and practice. If
$(c_{0},c_{1},\ldots,c_{n-1})\in{\bf C}$, then
$(c_{n-1},c_{0},\ldots,c_{n-2})\in{\bf C}$, this code ${\bf C}\subset{\bf
F}_{q}^{n}$ is called cyclic. The dual code of a cyclic code is a cyclic code.
A codeword ${\bf c}$ in a cyclic code is identified with a polynomial ${\bf
c}(x)=c_{0}+c_{1}x+\cdots+c_{n-1}x^{n-1}\in{\bf F}_{q}[x]/(x^{n}-1)$. Every
cyclic code is a principal ideal in the ring ${\bf F}_{q}[x]/(x^{n}-1)$ and
then generated by a factor $g$ of $x^{n}-1$. BCH codes were introduced in
1959-1960, see [3, 4, 19]. Let $n$ be a positive integer satisfying
$\gcd(n,q)=1$, and ${\bf Z}_{n}={\bf Z}/n{\bf Z}=\\{0,1,\ldots,n-1\\}$ be the
residue classes module $n$. A subset $C_{i}$ of ${\bf Z}_{n}$ is called a
$q$-cyclotomic coset if

$C_{i}=\\{i,iq,\ldots,iq^{l-1}\\},$

where $i\in{\bf Z}_{n}$ is fixed and $l$ is the smallest positive integer such
that $iq^{l}\equiv i$ $mod$ $n$. It is clear that $q$-cyclotomic cosets
correspond to irreducible factors of $x^{n}-1$ in ${\bf F}_{q}[x]$. A
generator polynomial of a cyclic code is the product of several irreducible
factors of $x^{n}-1$. The defining set of a cyclic code generated by $g$ is
the the following set

${\bf T}_{{\bf g}}=\\{i:g(\beta^{i})=0\\}.$

Hence the defining set of a cyclic code is the disjoint union of several
$q$-cyclotomic cosets. If there are $\delta-1$ consecutive elements in the
defining set of a cyclic code, the minimum distance of this cyclic code is at
least $\delta$. This is the BCH bound and $\delta$ is the designed distance,
see [3, 4, 19, 23]. There have been numerous papers on BCH codes, we refer to
the recent survey [12]. Some further bounds on minimum distances of cyclic
codes such as the Hartmann-Tzeng bound and the Roos bound were proposed and
used to construct good cyclic codes see [37, Chapter 6] and [21, Chapter 4].

Cyclic and negacyclic sum-rank codes in ${\bf F}_{q}^{s\times
s}\bigoplus\cdots\bigoplus{\bf F}_{q}^{s\times s}$ ($t$ copies) were
introduced in [9]. A sum-rank code ${\bf C}\subset{\bf F}_{q}^{s\times
s}\bigoplus\cdots\bigoplus{\bf F}_{q}^{s\times s}$ is called cyclic, if $({\bf
x}_{0},\ldots,{\bf x}_{t-1})\in{\bf C}$, then $({\bf x}_{t-1},{\bf
x}_{0},\ldots,{\bf x}_{t-2})\in{\bf C}$. A direct construction of cyclic and
negacyclic sum-rank codes from cyclic and negacyclic codes in the Hamming
metric was given in [9]. Cyclic skew cyclic sum-rank codes introduced in [28]
are special cases of cyclic sum-rank codes.

We identify the linear space ${\bf F}_{q}^{(n,n)}$ as the linear space of
$q$-polynomials,

${\bf
F}_{q}^{(n,n)}=\\{a_{0}x+a_{1}x^{q}+\cdots+a_{n-1}x^{q^{n-1}}:a_{i}\in{\bf
F}_{q^{n}},i=0,1,\ldots,n-1\\}.$

Let ${\bf C}_{0},\ldots,{\bf C}_{n-1}$ be linear $[t,k_{i},d_{i}]_{q^{n}}$
codes over ${\bf F}_{q^{n}}$. The sum-rank codes constructed in [7] is as
follows,

$SR({\bf C}_{0},\ldots,{\bf C}_{n-1})=\\{{\bf a}_{0}x+{\bf
a}_{1}x^{q}+\cdots+{\bf a}_{n-1}x^{q^{n-1}}:{\bf a}_{i}\in{\bf
C}_{i},i=0,1,\ldots,n-1\\}.$

Theorem 2.1 (see [7]). $SR({\bf C}_{0},\ldots,{\bf C}_{n-1})$ is a sum-rank
code with the block length $t$, the matrix size $n\times n$, the dimension
(over ${\bf F}_{q}$) $n(k_{0}+\cdots+k_{n-1})$ and the minimum sum-rank
distance at least

$d_{sr}\geq\min\\{d_{0},2d_{1},\ldots,nd_{n-1}\\}.$

When ${\bf C}_{0},\ldots,{\bf C}_{n-1}$ are cyclic codes, then $SR({\bf
C}_{0},\ldots,{\bf C}_{n-1})$ is a cyclic sum-rank code, see [9].

In the case of sum-rank codes in
${\mathbf{{F}}}_{q}^{(m,n)}\oplus{\mathbf{{F}}}_{q}^{(m,n)}$, where $n<m$. We
identify the matric space ${\bf F}_{q}^{(m,n)}$ as the space of all
$q$-polynomials $a_{n-1}\phi(x^{q^{n-1}})+\cdots+a_{0}\phi(x)$, where
$x\in{\mathbf{{F}}}_{q^{n}}$, $a_{m-1},\ldots,a_{0}\in{\bf F}_{q^{m}}$, and
$\phi:{\bf F}_{q^{n}}\longrightarrow{\bf F}_{q^{m}}$ is a
${\mathbf{{F}}}_{q}$-linear injective mapping. Let $C_{i}$ be linear
$[t,t-r_{i},d_{i}]_{q^{m}}$ codes, $i=0,1,\ldots,m-1$. Then we have a linear
sum-rank code

$SR({\bf C}_{0},\ldots,{\bf C}_{n-1})=\\{{\bf a}_{n-1}\phi(x^{q^{n-1}})+{\bf
a}_{n-2}\phi(x^{q^{n-2}})+\cdots+{\bf a}_{0}\phi(x),{\bf a}_{i}\in{\bf
C}_{i},i=0,1,\ldots,n-1\\}$

with the dimension (over ${\bf F}_{q^{m}}$)

$K=nt-r_{0}-\cdots-r_{n-1},$

and the minimum sun-rank distance

$d_{sr}\geq\min\\{d_{0},2d_{1},3d_{2},\ldots,nd_{n-1}\\}.$

This is the extension of the construction of sum-rank codes from codes in the
Hamming metric given in [7].

##  3 Distance-optimal cyclic codes with the minimum distance four

Let $q$ be a prime power satisfying $q\geq 4$ and $n=\frac{q^{m}-1}{\lambda}$,
where $m=1,2,3,\ldots$ are positive integers and $\lambda$ is a divisor of
$q^{m}-1$. Then each $q$-cyclotomic coset in ${\bf Z}_{n}$ has at most $m$
elements. The defining set

${\bf T}=C_{0}\bigcup C_{1}\bigcup C_{2}$

has at most $2m+1$ elements. An infinite family of cyclic
$[\frac{q^{m}-1}{\lambda},\geq\frac{q^{m}-1}{\lambda},\geq 4]_{q}$ codes is
constructed, since $0,1,2$ are in the defining set ${\bf T}$.

The volume of the ball of the radius $2$ in the Hamming metric space ${\bf
F}_{q}^{n}$ satisfies that

$V_{H}(q,2)>\frac{n(n-1)(q-1)^{2}}{2}.$

From the condition

$\lambda<\frac{q-1}{\sqrt{2q(1+\epsilon)}},$

and

$\lambda\leq q^{2},$

then

$\frac{n(n-1)(q-1)^{2}}{2}>n(n-1)(1+\epsilon)\geq q^{2m+1},$

when $m$ is sufficiently large. Then codes in this family are distance-
optimal, when $m$ is a sufficiently large positive integer.

Theorem 3.1. If $\lambda$ is a divisor of $q^{m}-1$ and satisfies
$\lambda<\frac{q-1}{\sqrt{2q(1+\epsilon)}}$, then a distance-optimal cyclic
$[\frac{q^{m}-1}{\lambda},\frac{q^{m}-1}{\lambda}-2m-1,4]_{q}$ code is
constructed, when $m$ is sufficiently large.

When $\lambda=1$, we always construct an infinite family of distance-optimal
cyclic $[q^{m}-1,q^{m}-2-2m,4]_{q}$ codes, for any prime power $q$. These
distance-optimal cyclic codes have the same parameters as distance-optimal
cyclic codes in [42, 11, 40].

For example, an infinite family of distance=optimal cyclic
$[\frac{5^{2m}-1}{3},\frac{5^{2m}-1}{3}-2m-1,4]_{25}$ codes can be obtained.
It seems this is a family of distance-optimal cyclic codes with new
parameters.

The Boston bound in [5, 43] asserts that if $0,1,3,5$ are in the defining set
of a $q$-ary cyclic code, then the minimum distance is at least four. Since
$3$ is in $C_{1}$ for a ternary cyclic code and $5$ is in $C_{1}$ for a
quinary cyclic code, we have the following result.

Theorem 3.2. A distance-optimal ternary cyclic $[3^{m}-1,3^{m}-2m-2,4]_{3}$
code with the defining set $C_{0}\bigcup C_{1}\bigcup C_{5}$ is constructed. A
distance-optimal quinary cyclic $[5^{m}-1,5^{m}-2m-2,4]_{3}$ code with the
defining set $C_{0}\bigcup C_{1}\bigcup C_{3}$ is constructed.

These two families of distance-optimal codes can be compared with distance-
optimal codes in [42, 11, 40].

##  4 Distance-optimal $q$-ary cyclic sum-rank codes with the minimum sum-rank
distance four

We need the following result on the ball of the radius $2$ in the sum-rank
metric space.

Lemma 4.1. There are $\frac{(q^{s_{1}}-1)(q^{s_{2}}-1)}{q-1}$ rank one
matrices in ${\mathbf{{F}}}_{q}^{(s_{1},s_{2})}$.

Proof. Each column can be a nonzero vector in ${\bf F}_{q}^{s_{1}}$, we have
$\frac{q^{s_{1}}-1}{q-1}$ such nonzero vectors. Moreover, at each column, we
can multiply an element in ${\bf F}_{q}$, this corresponds to a nonzero vector
in ${\bf F}_{q}^{s_{2}}$. Therefore there are
$\frac{(q^{s_{1}}-1)(q^{s_{2}}-1)}{q-1}$ rank one $s_{1}\times s_{2}$
matrices. The conclusion follows immediately.

Lemma 4.1. The volume of the ball $V_{sr}(q,2)\subset{\bf F}_{q}^{s\times
s}\bigoplus\cdots\bigoplus{\bf F}_{q}^{s\times s}$ satisfies

$V_{sr}(q,2)\geq\frac{t(t-1)(q^{s}-1)^{4}}{2(q-1)^{2}}.$

Proof. This conclusion follows from Lemma 4.1 directly.

Let the block length $t$ be $t=\frac{q^{sm}-1}{\lambda}$, where $\lambda$ is a
positive divisor of $q^{sm}-1$. The linear cyclic code ${\bf C}_{i}$ is the
trivial $[t,t,1]_{q^{s}}$ code, for $i=3,\ldots m-1$. The linear cyclic code
${\bf C}_{i}$ is the trivial cyclic $[t,t-1,2]_{q^{s}}$ code, for $i=1,2$. The
first code ${\bf C}_{0}$ is a linear cyclic $[t,t-2m-1,4]_{q^{s}}$ with the
defining set ${\bf T}=C_{0}\bigcup C_{1}\bigcup C_{2}$. Then the codimension
over ${\bf F}_{q^{s}}$ of this cyclic sum-rank code $SR({\bf
C}_{0},\ldots,C_{s-1})$ is $2m+3$.

Theorem 4.1. Let $q$ be a prime power, $s$ and $m$ be fixed positive integers.
Suppose that $\lambda$ is a positive divisor of $q^{sm}-1$ satisfying

$\lambda<\sqrt{\frac{(q^{s}-1)}{2(q-1)^{2}(1+\epsilon)}},$

where $\epsilon$ is an arbitrary small positive real number. Then we construct
a cyclic sum-rank code over ${\bf F}_{q}$ with the block length
$t=\frac{q^{sm}-1}{\lambda}$, the matrix size $s\times s$, the cardinality
$q^{s^{2}t-s(2m+3)}$ and the minimum sum-rank distance four. This cyclic sum-
rank code is distance-optimal.

Proof. From the condition on $\lambda$ and Lemma 4.2, it is easy to verify
that

$V_{sr}(q,2)>q^{s(2m+3)},$

when $m$ is sufficiently large. The conclusion follows immediately.

It is obvious that infinitely many families of distance-optimal cyclic sum-
rank codes can be obtained from Theorem 4.1.

Let $s_{1}<s_{2}$ and $m$ be a positive integers, and $\lambda$ be a divisor
of $q^{s_{2}m}-1$. We construct the block length
$t=\frac{q^{s_{2}m}-1}{\lambda}$, the matrix size $s_{1}\times s_{2}$
distance-optimal sum-rank codes with the minimum sum-rank distance $4$. The
linear cyclic code ${\bf C}_{i}$ is the trivial $[t,t,1]_{q^{s_{2}}}$ code,
for $i=3,\ldots s_{1}-1$. The linear cyclic code ${\bf C}_{i}$ is the trivial
cyclic $[t,t-1,2]_{q^{s_{2}}}$ code, for $i=1,2$. The first code ${\bf C}_{0}$
is a linear cyclic $[t,t-2m-1,4]_{q^{s_{2}}}$ with the defining set ${\bf
T}=C_{0}\bigcup C_{1}\bigcup C_{2}$. Then the codimension over ${\bf
F}_{q^{s_{2}}}$ of this cyclic sum-rank code $SR({\bf C}_{0},\ldots,C_{s-1})$
is $2m+3$.

Corollary 4.1. If the divisor $\lambda$ satisfies

$\lambda<\frac{q^{s_{1}}-1}{q-1}\sqrt{\frac{1}{2(1+\epsilon)q^{s_{2}}}},$

then an infinite family of distance-optimal codes of the block length $t$, the
matrix size $s_{1}\times s_{2}$ and the minimum distance $4$ is constructed.

We consider the packing density of the distance-optimal cyclic sum-rank code
${\bf C}$ with the block length $t=q^{sm}-1$, the matrix size $s\times s$, the
dimension $s^{2}t-s(2m+3)$ and the minimum distance four, constructed in
Theorem 4.1. The volume of the ball with the radius $1$ in the sum-rank metric
space ${\bf F}_{q}^{(s,s)}\oplus\cdots\oplus{\bf F}_{q}^{(s,s)}$ is

$V_{sr}(q,1)=1+\frac{(q^{s}-1)^{2}}{q-1}t.$

Then the packing density

$\mu({\bf
C})=\frac{V_{sr}(q,1)}{q^{s(2m+3)}}\approx\frac{1}{q^{sm+s+1}}\approx
O(\frac{1}{t}),$

when $q$ and $s$ are considered as fixed, $m$ goes to the infinity.

It is interesting to construct distance-optimal sum-rank codes with minimum
sum-rank distance four and larger packing densities.

##  5 Block length $q^{4}-1$ distance-optimal $q$-ary sum-rank codes with the
Singleton defect four

Let $q$ be a prime power and $n=q^{2}-1$. We consider the $q$-cyclotomic
cosets in ${\bf Z}_{n}$. Let the defining set be ${\bf T}=C_{0}\bigcup
C_{1}\bigcup C_{q+1}$. Then there are $4$ elements in this defining set. Let
${\bf C}$ be the cyclic code with this defining set. The four elements of this
defining set are $0,1,q,q+1$. Set ${\bf A}=\\{0,1\\}$ and ${\bf B}=\\{0,q\\}$,
then $\gcd(q,n)=1$. The condition of the Hartmann-Tzeng bound in [21, Theorem
4.5.6] is satisfied. Then the minimum distance $d({\bf C})$ is at least four.
This cyclic $[q^{2}-1,q^{2}-5,4]_{q}$ code is an almost MDS code.

Let the block length $t$ be $t=q^{4}-1$. The linear cyclic code ${\bf C}_{1}$
is the trivial $[t,t-1,2]_{q^{2}}$ code. The first code ${\bf C}_{0}$ is a
cyclic $[q^{4}-1,q^{4}-5,4]_{q^{2}}$ code constructed as above. Then the
codimension over ${\bf F}_{q^{2}}$ of this sum-rank code $SR({\bf
C}_{0},C_{1})$ with the matrix size $2\times 2$ is $5$.

Theorem 5.1 The $q$-ary and the matrix size $2\times 2$ sum-rank code $SR({\bf
C}_{0},{\bf C}_{1})$ is distance-optimal. The Singleton defect of this sum-
rank code is $4$.

Proof. We only need to prove that the volume $V_{sr}(q,2)$ of the ball with
the radius $2$ in the sum-rank metric space ${\bf
F}_{q}^{(2,2)}\oplus\cdots\oplus{\bf F}_{q}^{(2,2)}$ satisfies

$V_{sr}(q,2)>q^{10}.$

From Lemma 4.1, $V_{sr}(q,2)>\frac{t(t-1)(q^{2}-1)^{4}}{2(q-1)^{2}}>q^{13}$,
when $q\geq 3$. The first conclusion is proved. The Singleton defect is
$2(2t-4+1)-(4t-10)=4$. The second conclusion is proved.

The above sum-rank codes are cyclic sum-rank codes, see [9]. Infinitely many
distance-optimal sum-rank codes, which are close to Singleton-like bound and
have much larger $q^{4}-1>>q-1$ block lengths, are constructed. These sum-rank
codes are next best possibility to the almost MSRD codes with the Singleton
defect $2$. We do not know if these exists an such almost MSRD code.

##  6 Quasi-perfect $q$-ary sum-rank codes with the matrix size $2\times m$

We construct quasi-perfect $q$-ary sum-rank codes with the matrix size
$2\times m$ and the block length $t=\frac{q^{mu}-1}{q^{m}-1}$. The code ${\bf
C}_{1}$ is the trivial $[t,t-1,2]_{q^{m}}$ code. The code ${\bf C}_{0}$ is the
Hamming $[t,t-m,3]_{q^{m}}$ code. From a similar argument as Theorem 2.1, the
minimum sum-rank distance of $SR({\bf C}_{0},{\bf C}_{1})$ is at least $3$.
The codimension of this code over ${\bf F}_{q^{m}}$ is $m+1$.

Theorem 6.1. The sum-rank code $SR({\bf C}_{0},{\bf C}_{1})$ is distance-
optima and quasi-perfect.

Proof. The volume of the ball with the radius $2$ is at least

$V_{sr}(q,2)>\frac{t(t-1)}{2}\cdot(q+1)(q^{m}-1)^{2}>q^{m(u+1)}.$

Then these sum-rank codes are distance-optimal.

For any vector ${\bf v}$ in
${\mathbf{{F}}}_{q}^{(2,m)}\oplus\cdots\oplus{\mathbf{{F}}}_{q}^{(2,m)}$, it
can be represented by ${\bf v}={\bf v}_{1}x^{q}+{\bf v}_{0}x$, where ${\bf
v}_{i}\in{\mathbf{{F}}}_{q^{m}}^{t}$, $i=0,1$. We considered
${\mathbf{{C}}}_{1}$ as a covering code with the radius one. It is obvious
there is a codeword ${\mathbf{c}}_{1}\in{\mathbf{{C}}}_{1}$ such that
${\mathbf{v}}_{1}-{\mathbf{c}}_{1}$ has only one nonzero coordinate. This
nonzero coordinate can be at any position in $\\{0,1,\ldots,t-1\\}$. On the
other hand, it is well-known that the Hamming code ${\mathbf{{C}}}_{0}$ is a
perfect code in the Hamming metric. For the vector ${\mathbf{v}}_{0}$ there is
a codeword ${\mathbf{c}}_{0}\in{\mathbf{{C}}}_{0}$ such that
${\mathbf{v}}_{0}-{\mathbf{c}}_{0}$ has only one nonzero coordinate. This
nonzero coordinate has a special coordinate position. Then by choosing
${\mathbf{c}}_{1}$ suitably, we find a codeword ${\mathbf{c}}\in
SR({\mathbf{{C}}}_{0},{\mathbf{{C}}}_{1})$ such that
${\mathbf{v}}-{\mathbf{c}}$ has only one nonzero coordinate position. Then the
sum-rank weight of ${\mathbf{v}}-{\mathbf{c}}$ is at most $2$. Then the
covering radius of the sum-rank code
$SR({\mathbf{{C}}}_{0},{\mathbf{{C}}}_{1})$ is at most two. It is a quasi-
perfect sum-rank code.

##  7 Quasi-perfect binary sum-rank codes with the matrix size $2\times 2$ and
the minimum distance four

We recall the following result in [16]. Let
$s_{m,q}=3(q^{\lfloor\frac{m-3}{2}\rfloor}+q^{\lfloor\frac{m-3}{2}\rfloor-1}+\ldots+q)+2$
and $n=2q^{\frac{m-2}{2}}+s_{m,q}$. Let $q$ be an even square, $m\geq 7$ be
odd. An infinite family of quasi-perfect $[n,n-m,4]_{q}$ codes was
constructed, see [16, Proposition 2.5]. Other infinite families of quasi-
perfect $[n,n-m,4]_{q}$ codes was constructed in [16, Propositon 4.1], for
even prime power $q$. We need the following lemma for binary sum-rank codes of
the matrix size $2\times 2$.

Lemma 7.1. Let ${\bf C}_{0}\subset{\bf F}_{4}^{t}$ and ${\bf C}_{1}\subset{\bf
F}_{4}^{t}$ be two linear $[t,k_{0},d_{0}]_{4}$ and $[t,k_{1},d_{1}]_{4}$
codes over ${\bf F}_{4}$. Then a block length $t$ and matrix size $2\times 2$
binary linear sum-rank-metric code $SR({\bf C}_{0},{\bf C}_{1})$ can be
constructed explicitly. The minimum sum-rank distance of $SR({\bf C}_{0},{\bf
C}_{1})$ is at least $\min\\{d_{0},2d_{1}\\}$. The dimension of $SR({\bf
C}_{0},{\bf C}_{1})$ over ${\bf F}_{2}$ is $\dim_{{\bf F}_{2}}(SR({\bf
C}_{0},{\bf C}_{1}))=2(k_{0}+k_{1})$. Moreover for any two codewords, ${\bf
a}_{0}\in{\bf C}_{0}$ and ${\bf a}_{1}\in{\bf C}_{1}$, set

$I=supp({\bf a}_{0})\cap supp({\bf a}_{1}),$

then

$wt_{sr}({\bf a}_{1}x+{\bf a}_{0}x^{2})=2wt_{H}({\bf a}_{0})+2wt_{H}({\bf
a}_{1})-3|I|.$

Proof. For the coordinate position $i\in supp({\bf a}_{0})\backslash supp({\bf
a}_{1})$, it is clear the matrix ${\bf a}_{1}x+{\bf a}_{0}x^{2}$ in this
position is of the rank $2$. For the coordinate position $i\in supp({\bf
a}_{1})\backslash supp({\bf a}_{0})$, it is clear the matrix ${\bf
a}_{1}x+{\bf a}_{0}x^{2}$ in this position is of the rank $2$. For these
coordinate positions $i\in I$, it is clear that the matrix is of rank $1$,
since there is one nonzero root $x=\frac{a_{i,1}}{a_{i,0}}$, where

${\bf a}_{0}=(a_{1,0},\ldots,a_{t,0}),$ ${\bf
a}_{1}=(a_{1,1},\ldots,a_{t,1}).$

Then it follows that

$wt_{sr}({\bf a}_{1}x+{\bf a}_{0}x^{2})=2(wt_{H}({\bf
a}_{0})-|I|)+2(wt_{H}({\bf a}_{1})-|I|)+|I|.$

The conclusion follows immediately.

From Lemma 7.1, for two vectors ${\bf a}_{0}$ and ${\bf a}_{1}$ in
${\mathbf{{F}}}_{4}^{t}$ satisfying

$supp({\bf a}_{0})=supp({\bf a}_{1}),$ $wt_{sr}({\bf a}_{1}x+{\bf
a}_{0}x^{2})=wt({\bf a}_{0})=wt({\bf a}_{1}).$

From the following result, quasi-perfect binary sum-rank codes of the matrix
size $2\times 2$ can be obtained.

Theorem 7.1. Let ${\bf C}_{1}\subset{\bf F}_{4}^{t}$ be the linear
$[t,t-1,2]_{4}$ and $[t,k_{2},4]_{4}$ code and and ${\bf C}_{0}\subset{\bf
F}_{4}^{t}$ be a $[t,k,4]_{4}$ code with the covering radius two. Then a block
length $t$ and matrix size $2\times 2$ binary linear sum-rank-metric code
$SR({\bf C}_{0},{\bf C}_{1})$ with the minimum sum-rank distance four is
constructed explicitly. The covering radius of this binary sum-rank code is
two. Then $SR({\bf C}_{0},{\bf C}_{1})$ is quasi-perfect..

Proof. For any vector ${\bf v}$ in
${\mathbf{{F}}}_{4}^{(2,2)}\oplus\cdots\oplus{\mathbf{{F}}}_{q}^{(2,2)}$ ($t$
copies), it can be represented by ${\bf v}={\bf v}_{1}x^{2}+{\bf v}_{0}x$,
where ${\bf v}_{i}\in{\mathbf{{F}}}_{4}^{t}$, $i=0,1$. We considered
${\mathbf{{C}}}_{1}$ as a covering code with the radius one. It is obvious
there is a codeword ${\mathbf{c}}_{1}\in{\mathbf{{C}}}_{1}$ such that
${\mathbf{v}}_{1}-{\mathbf{c}}_{1}$ has exactly two nonzero coordinates. These
two nonzero coordinates can be at any two different positions in
$\\{0,1,\ldots,t-1\\}$. On the other hand, the quasi-perfect code
${\mathbf{{C}}}_{0}$ has the covering radius two in the Hamming metric. For
the vector ${\mathbf{v}}_{2}$ there is a codeword
${\mathbf{c}}_{0}\in{\mathbf{{C}}}_{0}$ such that
${\mathbf{v}}_{0}-{\mathbf{c}}_{0}$ has at most two nonzero coordinates. These
two nonzero coordinates have special coordinate positions. Then by choosing
${\mathbf{c}}_{1}$ suitably, we find a codeword ${\mathbf{c}}\in
SR({\mathbf{{C}}}_{0},{\mathbf{{C}}}_{1})$ such that
${\mathbf{v}}-{\mathbf{c}}$ has at most two nonzero coordinate positions. Then
the sum-rank weight of ${\mathbf{v}}-{\mathbf{c}}$ is at most $2$. Then the
covering radius of the binary sum-rank code
$SR({\mathbf{{C}}}_{0},{\mathbf{{C}}}_{1})$ is at most two. It is a quasi-
perfect sum-rank code.

From Theorem 7.1, many quasi-perfect binary sum-rank codes with the matrix
size $2\times 2$ and the minimum sum-rank distance four can be constructed
explicitly, from these quasi-perfect codes over ${\mathbf{{F}}}_{4}$
constructed in [16].

##  8 Plotkin sum of sum-rank codes

As in the case of the matrix size $1\times 1$ sum-rank codes (codes in the
Hamming metric), the Plotkin sum of sum-rank codes of the matrix size $n\times
m$, $n\leq m$, can be defined and nice sum-rank codes can be obtained.

Let ${\bf C}_{1}$ and ${\bf C}_{2}$ be two sum-rank codes in
${\mathbf{{F}}}_{q}^{(n,m)}\oplus\cdots\oplus{\mathbf{{F}}}_{q}^{(n,m)}$, with
minimum sum-rank distances $d_{1}$, $d_{2}$, and dimensions $k_{1}$ and
$k_{2}$. Then we define their Plotkin sum-rank code as

${\mathrm{Plotkin}}({\bf C}_{1},{\bf C}_{2})=\\{({\bf c}_{1}|{\bf c}_{1}+{\bf
c}_{2}):{\mathbf{c}}_{1}\in{\mathbf{{C}}}_{1},{\mathbf{c}}_{2}\in{\mathbf{{C}}}_{2}\\}.$

Theorem 8.1. The dimension of the Plotkin sum is $k_{1}+k_{2}$ and the minimum
sum-rank distance the Plotkin sum is $\min\\{2d_{1},d_{2}\\}$.

Proof. If ${\bf c}_{2}$ is not zero,

$wt_{sr}({\bf c}_{1}|{\bf c}_{1}+{\bf
c}_{2})=wt_{sr}({\mathbf{c}}_{1})+wt_{sr}({\mathbf{c}}_{1}+{\mathbf{c}}_{2})\geq
wt_{sr}({\mathbf{c}}_{1}+({\mathbf{c}}_{1}+{\mathbf{c}}_{2}))=wt_{sr}({\mathbf{c}}_{2})\geq
d_{2}.$

If ${\mathbf{c}}_{2}=0$, then

$wt_{sr}({\bf c}_{1}|{\bf c}_{1})=2wt_{sr}({\mathbf{c}}_{1})\geq 2d_{1}.$

The conclusion follows immediately.

From Theorem 2.1, we get the block length $t$ sum-rank code ${\bf C}_{1}$ with
the matrix size $s\times$ and the minimum sum-rank distance $2$ and the
codimension $s$ directly, for an arbitrary positive integer $t$. From Theorem
4.1, we get an infinite family of distance-optimal binary sum-rank codes
${\mathbf{{C}}}_{2}$ with the block length $t=2^{sm}-1$, the matrix size
$s\times s$, the codimension $2m+3$ (over ${\mathbf{{F}}}_{2^{s}}$), and the
minimum sum-rank distance four.

Then we can obtain new distance-optimal binary sum-rank codes with the block
length $2t=2(2^{sm}-1)$.

Corollary 8.1. Let $s$ and $m$ be two positive integers. Then the Plotkin sum
${\mathrm{Plotkin}}({\mathbf{{C}}}_{1},{\mathbf{{C}}}_{2})$ is another
distance-optimal binary sum-rank codes with the block length $2t$.

Proof. The minimum distance of
${\mathrm{Plotkin}}({\mathbf{{C}}}_{1},{\mathbf{{C}}}_{2})$ is four.

$V_{sr}(2,2)>\frac{2t(2t-1)}{2}\cdot(2^{s}-1)^{2}=2(2^{sm}-1)(2^{sm}-\frac{3}{2})(2^{s}-1)^{2}\geq
2^{s(2m+4)},$

when $m$ is large. The conclusion follows directly.

##  9 Conclusion

To construct distance-optimal and quasi-perfect codes in the sum-rank metric
are interesting and challenging problems. In this paper, infinite many
families of distance-optimal cyclic sum-rank codes with the matrix size
$2\times 2$ and the minimum sum-rank distance four were constructed. An
infinite family of long distance-optimal $q$-ary sum-rank codes with the block
length $q^{4}-1$, the matrix size $2\times 2$, the minimum sum-rank distance
four and the Singleton defect four was given. Moreover, several families of
quasi-perfect sum-rank codes with the matrix size $n\times m$, $2\leq n\leq
m$, were also presented. We also showed that the Plotkin sum of sum-rank codes
gives us more distance-optimal binary sum-rank codes.

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