# Relating the Roe algebra of a space to the uniform Roe algebras of its
discretizations

V. Manuilov  Moscow Center for Fundamental and Applied Mathematics, Moscow
State University, Leninskie Gory 1, Moscow, 119991, Russia
<EMAIL_ADDRESS>
###### Abstract.

The Roe algebra $C^{*}(X)$ is a non-commutative $C^{*}$-algebra reflecting
metric properties of a space $X$, and it is interesting to understand relation
between the Roe algebra of $X$ and the uniform Roe algebra of its
discretizations. Here we construct, for a simplicial space $X$, a continuous
field of $C^{*}$-algebras over $\mathbb{N}\cup\\{\infty\\}$ with the fibers
over finite points the uniform $C^{*}$-algebras of discretizations of $X$, and
the fiber over $\infty$ the Roe algebra of $X$. We also construct the direct
limit of the uniform Roe algebras of discretizations and its embedding into
the Roe algebra of $X$.

##  1\. Introduction

Roe algebras play an increasingly important role in the index theory of
elliptic operators on noncompact manifolds and their generalizations [10, 6,
8]. Following the ideology of noncommutative geometry [2], they provide an
interplay between metric spaces (e.g. manifolds) and (noncommutative)
$C^{*}$-algebras.

Recall the definition of the Roe algebra of a metric space $X$ [10]. Let
$H_{X}$ be a Hilbert space with an action of the algebra $C_{0}(X)$ of
continuous functions on $X$ vanishing at infinity (i.e. a $*$-homomorphism
$\pi:C_{0}(X)\to\mathbb{B}(H_{X})$). For a Hilbert space $H$ we write
$\mathbb{B}(H)$ (resp., $\mathbb{K}(H)$) for the algebra of all bounded
(resp., all compact) operators on $H$. We will assume that

$\\{\pi(f)\xi:f\in C_{0}(X),\xi\in H_{X}\\}\quad\mbox{is\ dense\ in}\quad
H_{X}$ (1)

and that

$\pi(f)\in\mathbb{K}(H_{X})\quad\mbox{implies\ that}\quad f=0.$ (2)

An operator $T\in\mathbb{B}(H_{X})$ is locally compact if the operators
$T\pi(f)$ and $\pi(f)T$ are compact for any $f\in C_{0}(X)$. It has finite
propagation if there exists some $R>0$ such that $\pi(f)T\pi(g)=0$ whenever
the distance between the supports of $f,g\in C_{0}(X)$ is greater than $R$.
The Roe algebra $C^{*}(X,H_{X})$ is the norm completion of the $*$-algebra of
locally compact, finite propagation operators on $H_{X}$. As it does not
depend on the choice of $H_{X}$ satisfying (1) and (2), it is usually denoted
by $C^{*}(X)$. It is convenient to have a Borel measure on $X$. When the
measure $\mu$ on $X$ has no atoms, the simplest choice $H_{X}=L^{2}(X)$ with
the standard action of $C_{0}(X)$ satisfies the properties (1) and (2), so it
suffices to use $L^{2}(X)$ to define the Roe algebra.

When $X$ is discrete (with an atomic measure), the choice $H_{X}=l^{2}(X)$
does not satisfy the condition (2). In order to fix this, one may take
$H_{X}=l^{2}(X)\otimes H$ with an infinitedimensional Hilbert space $H$ to
obtain the Roe algebra. But there is also another option: still to use
$H_{X}=l^{2}(X)$. The resulting $C^{*}$-algebra is called the uniform Roe
algebra of $X$, and is denoted by $C^{*}_{u}(X)$. This $C^{*}$-algebra differs
from the Roe algebra $C^{*}(X)$. It is more tractable, but has less relations
with elliptic theory.

Manifolds and some other spaces $X$ are often endowed with discrete subspaces
$D\subset X$ that are $\varepsilon$-dense for some $\varepsilon$, e.g.
lattices in Lie groups, or, more generally, Delone sets in metric spaces [1].
Some problems related to $X$ may become simpler when reduced to such
discretization $D$. In particular, it would be interesting to understand
relation between the Roe algebra of $X$ and the uniform Roe algebras of its
discretizations $D$. The latter are embedded into $C^{*}(X)$, and in [7] we
have shown that a large class of non-trivial projections in $C^{*}_{u}(D)$
vanish in the $K$-theory of $C^{*}(X)$. Delone sets on general metric spaces
are too vague to establish such relation, so we restrict ourselves to the case
when $X$ is a simplicial complex. The set of vertices is a discretization of
$X$, and in order to get more dense discretizations we can divide the
simplices of $X$ into smaller simplices.

In this paper we construct a continuous field of $C^{*}$-algebras over
$\mathbb{N}\cup\\{\infty\\}$ such that the fiber over $\infty$ is the Roe
algebra of $X$, while the fiber over any finite point is the uniform Roe
algebra of a certain discretization $D(n)$, $n\in\mathbb{N}$, of $X$. Such non
locally trivial continuous fields of $C^{*}$-algebras are interesting because
they provide relations between fibers over different points. In particular,
they may provide a map from the $K$-theory group of the fiber over $\infty$ to
the $K$-theory group of the fibers over finite points (cf. [4]). In [9] we
have implemented a similar construction of a continuous field in the case
$X=\mathbb{R}$ using the group structure on $\mathbb{R}$.

We also construct the direct limit of the uniform Roe algebras of
discretizations $D(n)$ of $X$ that become more and more dense, and an
embedding of this direct limit into $C^{*}(X)$. Such direct limits are good
candidates for what could be the uniform Roe algebra of a non-discrete space.

Let $X$ be a countable simplicial complex consisting of simplices
$\Delta_{\alpha}^{k}$ of dimension $k$. We assume that $X$ is uniformly
locally bounded, i.e. that each simplex has a uniformly bounded number of
adjacent simplices, and that $X$ is finitedimensional, i.e. that all
dimensions of simplices are uniformly bounded. We say that a simplex has
maximal dimension if it is not a subset of another simplex ($X$ may have
simplices of different maximal dimensions). We endow $X$ with a metric $d$
(resp., a Borel measure $\mu$) such that the restriction of $d$ (resp., of
$\mu$) onto each simplex $\Delta^{k}_{\alpha}$ (resp. onto each simplex of
maximal dimension) coincides with the standard metric (resp., standard
measure) on the standard simplex

$\Delta=\Delta(1)=\\{(x_{0},x_{1},\ldots,x_{k})\in\mathbb{R}^{k+1}:\sum\nolimits_{j=0}^{k}x_{i}=1;x_{j}\geq
0,j=0,1,\ldots,k\\}.$ (3)

If $x,y\in X$ do not lie in one simplex then $d(x,y)$ is defined as the
infimum of lengths of paths connecting $x$ and $y$. After fixing the isometric
homeomorphisms of the simplices of $X$ with the standard simplex we shall use
the coordinates on the standard simplex as local coordinates on $X$.

##  2\. Discretizations of $X$ and partitions of unity

For the given simplicial structure of $X$, let $D=D(1)$ denote the discrete
metric space consisting of all the vertices $\Delta^{0}_{\alpha}$, $\alpha\in
A$. It is a 1-net for $X$. The assumption that $X$ is finitedimensional and
uniformly locally bounded implies that $D(1)$ has bounded geometry, i.e. the
number of points of any ball of radius $R$ in $D(1)$ is bounded by some number
$b(R)$ which depends only on $R$.

Consider a subdivision of the simplicial structure
$\\{\Delta^{k}_{\alpha}\\}_{\alpha\in
A}=\\{\Delta^{k}_{\alpha_{1}}(1)\\}_{\alpha_{1}\in A_{1}}$ of $X$ into smaller
simplices $\Delta^{k}_{\alpha_{2}}(2)$, $\alpha_{2}\in A_{2}$, such that each
simplex $\Delta^{k}_{\alpha_{1}}$ is the union of $2^{k}$ simplices
$\Delta^{k}_{\alpha_{2}}(2)$, $\alpha_{2}\in A_{2}$, isometric to the standard
simplex

$\Delta(2)=\\{(x_{0},x_{1},\ldots,x_{k})\in\mathbb{R}^{k+1}:\sum\nolimits_{j=0}^{k}x_{j}=\frac{1}{2};x_{j}\geq
0,j=0,1,\ldots,k\\}.$

Let $D(2)$ be the metric space of all vertices of the new simplicial structure
$\\{\Delta^{k}_{\alpha_{2}}(2)\\}_{\alpha_{2}\in A_{2}}$.

Proceeding, we obtain simplicial structures
$\\{\Delta^{k}_{\alpha_{n}}(n)\\}_{\alpha_{n}\in A_{n}}$ such that each
$\Delta^{k}_{\alpha_{n}}(n)$ is isometric to the standard simplex

$\Delta(n)=\\{(x_{0},x_{1},\ldots,x_{k})\in\mathbb{R}^{k+1}:\sum\nolimits_{j=0}^{k}x_{j}=\frac{1}{2^{n}};x_{j}\geq
0,j=0,1,\ldots,k\\}.$

We also obtain a sequence of discretizations $D(1)\subset
D(2)\subset\cdots\subset X$ such that $D(n)$ is an $\frac{1}{2^{n}}$-net in
$X$. The metric spaces $D(n)$ are also of bounded geometry for each
$n\in\mathbb{N}$.

For each vertex $u=\Delta^{0}_{\alpha_{n}}(n)\in D(n)$ let
$\operatorname{St}_{u}\subset X$ denote the open star of $u$. Then
$\\{\operatorname{St}_{u}:u\in D(n)\\}$ is an open cover of $X$. Let
$\\{\varphi^{n}_{u}:u\in D(n)\\}$ be the piecewise linear partition of unity
for this cover. If $u\in\Delta_{\alpha_{n}}^{k}(n)\subset D(n)$ has
coordinates $(\frac{1}{2^{n}},0,\ldots,0)$ in the standard simplex then
$\varphi^{n}_{u}(x)=x_{0}$ for any $x\in\Delta(n)$.

Let $G=(g_{uv})_{u,v\in D(n)}$, where
$g_{uv}=\langle\varphi^{n}_{u},\varphi^{n}_{v}\rangle$, be the Gram matrix for
the set of functions $\\{\varphi^{n}_{u}\\}_{u\in D(n)}$. With an abuse of
notation we shall use the same $G$ to denote the operator on the Hilbert space
$l^{2}(D(n))$ with the matrix $G$ with respect to the standard basis of
$l^{2}(D(n))$, which consists of the functions $\delta_{u}$, $u\in D(n)$,
given by $\delta_{u}(v)=\left\\{\begin{array}[]{cl}1,&\mbox{if\ }v=u;\\\
0,&\mbox{if\ }v\neq u.\end{array}\right.$ Clearly, the operator $G$ is
positive.

######  Theorem 1.

There exists $C>0$ such that the operator $G$ on $l^{2}(D(n))$ is bounded from
below by $\frac{C}{2^{n}}$.

###### Proof.

It suffices to prove the claim for $n=1$, as the general case can be obtained
by scaling. We write $\Delta$ for $\Delta(1)$ and $D$ for $D(1)$ to simplify
the notation.

Let $\\{\Delta_{\beta}\\}$ be the set of simplices of maximal dimension. Note
that

$g_{uv}=\langle\varphi_{u},\varphi_{v}\rangle=\int_{X}\bar{\varphi}_{u}(x)\varphi_{v}(x)d\mu(x)=\sum\nolimits_{\beta}\int_{\Delta_{\beta}}\bar{\varphi}_{u}(x)\varphi_{v}(x)d\mu(x)=\sum\nolimits_{\beta}g^{\beta}_{uv},$

$u,v\in D$. Let $G^{\beta}$ be the operator on $l^{2}(D)$ with the matrix
$(g^{\beta}_{uv})_{u,v\in D}$ with respect to the standard basis. Then
$G=\sum_{\beta}G^{\beta}$, where the sum is $*$-stronlgy convergent due to
finite propagation of $G$. For $\Delta_{\beta}$ let $u_{0},u_{1},\ldots,u_{i}$
be the set of its vertices, and let $q_{\beta}$ denote the projection in
$l^{2}(D)$ onto the linear span of $\delta_{j}$, $j=0,1,\ldots,i$.

We claim that there exists $C>0$ such that $G^{\beta}\geq Cq_{\beta}$.
Assuming that this claim holds, it remains to note that each vertex lies in at
least one simplex of maximal dimension, hence $\sum_{\beta}q_{\beta}\geq 1$,
from which it follows that $G=\sum_{\beta}G^{\beta}\geq
C\sum_{\beta}q_{\beta}\geq C\cdot 1$.

Fix a simplex $\Delta^{k}_{\beta}$ of maximal dimension $k$, with vertices
$u_{0},u_{1},\ldots,u_{k}$. After identifying the simplex with the standard
simplex (3), these vertices have coordinates
$(1,0,\ldots,0),\ldots,(0,0,\ldots,1)$, and the functions $\varphi_{u_{i}}$
are identified with the functions $x_{i}$ on the standard simplex. Therefore,

$g^{\beta}_{u_{i}u_{j}}=\int_{\Delta_{\beta}^{k}}x_{i}x_{j}d\mu(x).$

By the definition of the projection $q_{\beta}$, the finite rank operator
$q_{\beta}G^{\beta}q_{\beta}$ has $(g^{\beta}_{u_{i}u_{j}})_{i,j=0}^{k}$ as
its matrix.

It remains to show that the positive operator $q_{\beta}G^{\beta}q_{\beta}$ is
separated from zero. Fix a $k$-dimensional cube $K$ with the sides of length
$a$, centered at the center $\frac{1}{k+1}(1,1,\ldots,1)$ of the simplex
$\Delta^{k}$. If $a$ is sufficiently small, $K\subset\Delta^{k}$. Then

$g^{\beta}_{u_{i}u_{j}}=\int_{K}x_{i}x_{j}d\mu(x)+\int_{\Delta^{k}\setminus
K}x_{i}x_{j}d\mu(x)=g^{\prime}_{u_{i}u_{j}}+g^{\prime\prime}_{u_{i}u_{j}},$

and both matrices, $(g^{\prime}_{u_{i}u_{j}})_{i,j=0}^{k}$ and
$(g^{\prime\prime}_{u_{i}u_{j}})_{i,j=0}^{k}$ are positive, so it would
suffice to prove that the first one is separated from zero.

Let us calculate $g^{\prime}_{u_{i}u_{j}}=\int_{K}x_{i}x_{j}d\mu(x)$. Let $U$
be an orthogonal operator on $\mathbb{R}^{k+1}$ that moves the standard
simplex to a subset of the hyperplane $x_{0}=\frac{1}{\sqrt{k+1}}$ (and the
center of this simplex to the point $(\frac{1}{\sqrt{k+1}},0,\ldots,0)$).
Changing variables $Ux=y$, we get

$(g^{\prime}_{u_{i}u_{j}})_{i,j=0}^{k}=\int_{K}(x_{i}x_{j})_{i,j=0}^{k}d\mu(x)=U\left(\int_{U(K)}(y_{p}y_{q})_{p,q=0}^{k}d\mu(y)\right)U^{*},$

so it remains to separate the matrix

$\int_{U(K)}(y_{p}y_{q})_{p,q=0}^{k}d\mu(y)$ (4)

from zero.

If $p\neq q$ then

$\int_{U(K)}y_{p}y_{q}d\mu(y)=\int_{-a}^{a}\cdots\int_{-a}^{a}y_{p}y_{q}\,dy_{1}\cdots
dy_{k}=0,$

so the matrix (4) is diagonal, and it remains to calculate

$\int_{U(K)}y_{p}^{2}d\mu(y)=\int_{-a}^{a}\cdots\int_{-a}^{a}y^{2}_{p}\,dy_{1}\cdots
dy_{k},$

which, for $p=0$, equals $(2a)^{k}\frac{1}{k+1}$, while for $p>0$, it equals
$(2a)^{k-1}\frac{2a^{3}}{3}$. If $C$ is smaller than both these two numbers
then $\int_{U(K)}(y_{p}y_{q})_{p,q=0}^{k}d\mu(y)\geq CI$, where $I$ denotes
the unit matrix. ∎

######  Corollary 2.

The operator $G$ is bounded, invertible and has finite propagation.

###### Proof.

Invertibility follows from Theorem 1. Finite propagation follows from the fact
that if $u,v\in D(n)$ and if
$\operatorname{St}_{u}\cap\operatorname{St}_{v}=\emptyset$ then
$\langle\varphi_{u},\varphi_{v}\rangle=0$, so uniform local finiteness of $X$
implies that $G$ has finite propagation. Finally, as all matrix entries are
uniformly bounded, an operator of finite propagation is bounded. ∎

##  3\. Two maps

In this section we construct the maps $\alpha_{n}:C^{*}_{u}(D(n))\to C^{*}(X)$
and $\beta_{n}:C^{*}(X)\to C^{*}_{u}(D(n))$, $n\in\mathbb{N}$. Let $p_{n}$
denote the projection, in $L^{2}(X)$, onto the closure $H_{n}$ of the linear
span of $\varphi^{n}_{x}$, $x\in D(n)$. The construction is the same for each
$n$, so we shall skip $n$ from the notation and write $D$ for $D(n)$, $H$ for
$H_{n}$, and $\varphi_{u}$ for $\varphi^{n}_{u}$.

Set $C=G^{-1/2}$, $C=(c_{uv})_{u,v\in D}$, where $G$ is the operator
determined by the Gram matrix for $\\{\varphi_{u}\\}$, $u\in D$. By functional
calculus, $C$ can be approximated by polynomials in $G$, hence $C$ lies in the
norm closure of operators of finite propagation, i.e. $C\in C^{*}_{u}(D)$.

Set $\psi_{v}=\sum_{u\in D}c_{uv}\varphi_{u}$. Then

$\displaystyle\langle\psi_{u},\psi_{v}\rangle$ $\displaystyle=$
$\displaystyle\Bigl{\langle}\sum\nolimits_{x\in
D}c_{xu}\varphi_{x},\sum\nolimits_{y\in
D}c_{yv}\varphi_{y}\Bigr{\rangle}=\sum\nolimits_{x,y\in
D}\bar{c}_{xu}c_{yv}\langle\varphi_{x},\varphi_{y}\rangle$ $\displaystyle=$
$\displaystyle\sum\nolimits_{x,y\in
D}\bar{c}_{xu}c_{yv}g_{xy}=(CGC)_{uv}=\delta_{uv},$

hence $\\{\psi_{u}\\}_{u\in D}$ is an orthonormal system. Invertibility of $C$
implies that the closures of the linear spans of $\\{\varphi_{u}\\}_{u\in D}$
and of $\\{\psi_{u}\\}_{u\in D}$ coincide. The advantage of this orthonormal
system with respect to the system obtained from $\\{\varphi_{u}\\}_{u\in D}$
by the Gram–Schmidt orthogonalization is that it is obtained from the original
non-orthogonal system by using an operator from $C^{*}_{u}(\mathbb{Z})$.

Consider one more basis for $H$. By construction of $C$, for any
$\varepsilon>0$ there exists an operator
$C_{\varepsilon}\in\mathbb{B}(l^{2}(D))$ of finite propagation
$M_{\varepsilon}$, with the matrix $C_{\varepsilon}=(c^{\varepsilon}_{uv})$,
such that $\|C-C_{\varepsilon}\|<\varepsilon$. Set
$\psi_{v}^{\varepsilon}=\sum_{u\in D}c_{uv}^{\varepsilon}\varphi_{u}$. The
basis $\\{\psi^{\varepsilon}_{v}\\}_{v\in D}$ is only almost orthonormal, but,
unlike $\\{\psi_{v}\\}_{v\in D}$, it is compactly localized in $D$.

Let $U:l^{2}(D)\to L^{2}(X)$ be the isometry determined by
$U(\delta_{u})=\psi_{u}$, $u\in D$. Then the range of $U$ is $H\subset
L^{2}(X)$. Define a map $\alpha:C^{*}_{u}(D)\to C^{*}(X)$ by
$\alpha(T)(\xi)=UTU^{*}(p\xi)$, where $\xi\in L^{2}(X)$. Note that if
$T=(t_{uv})_{u,v\in D}$ is the matrix of $T\in\mathbb{B}(l^{2}(D))$ then

$\alpha(T)(\xi)=\sum\nolimits_{u,v\in
D}t_{uv}\psi_{u}\langle\psi_{v},\xi\rangle,\quad\xi\in L^{2}(X).$

Clearly, $\alpha$ is a $*$-homomorphism, in particular, it is isometric. As
$T$ is bounded, $\alpha(T)$ is bounded as well.

Let also $U_{\varepsilon}:l^{2}(D)\to L^{2}(X)$ be an operator given by
$U_{\varepsilon}\delta_{u}=\psi^{\varepsilon}_{u}$, $u\in D$.

For a matrix $A=(a_{uv})_{u,v\in D}$ define an operator $\gamma(A)$ on $H$ by
$\gamma(A)\varphi_{v}=\sum_{u\in D}a_{uv}\varphi_{u}$. Then one can write
$\alpha(T)$ as $\alpha(T)=\gamma(C^{-1}TC)$.

######  Lemma 3.

The map $\gamma$ is bounded.

###### Proof.

Let $f=\sum_{u\in D}f_{u}\varphi_{u}\in H$. Then, by Corollary 2, the norm
$\|f\|^{2}_{L^{2}(X)}=\sum_{u,v\in D}g_{uv}\bar{f}_{u}f_{v}$ is equivalent to
the norm $\|f\|^{2}_{2}=\sum_{u\in D}|f_{u}|^{2}$. Therefore, there exists
$K>0$ such that

$\displaystyle\|\gamma(A)f\|_{L^{2}(X)}\leq
K\|\gamma(A)f\|_{2}=K\|A\xi\|_{2}\leq K\|A\|\|\xi\|_{2}\leq
K^{2}\|A\|\|f\|_{L^{2}(X)},$

where $\xi=\sum_{u\in D}f_{u}\delta_{u}\in l^{2}(D)$. ∎

Set $\alpha_{\varepsilon}(T)=\gamma(C_{\varepsilon}^{-1}TC_{\varepsilon})$.

######  Lemma 4.

For sufficiently small $\varepsilon$ there exists $M>0$ such that
$\|\alpha(T)-\alpha_{\varepsilon}(T)\|<M\varepsilon$.

###### Proof.

One should take $\varepsilon$ small enough to provide invertibility of
$\gamma(C_{\varepsilon})$. Then

$\displaystyle\|\alpha(T)-\alpha_{\varepsilon}(T)\|$ $\displaystyle\leq$
$\displaystyle\|\gamma(C^{-1})-\gamma(C_{\varepsilon}^{-1})\|\cdot\|\gamma(T)\|\cdot\|\gamma(C)\|$
$\displaystyle+\|\gamma(C_{\varepsilon}^{-1})\|\cdot\|\gamma(T)\|\cdot\|\gamma(C)-\gamma(C_{\varepsilon})\|$
$\displaystyle\leq$
$\displaystyle\|\gamma(C-C_{\varepsilon})\|\cdot\|\gamma(C^{-1})\|\cdot\|\gamma(C_{\varepsilon}^{-1})\cdot\|\gamma(T)\|\|\cdot\|\gamma(C)\|$
$\displaystyle+\|\gamma(C_{\varepsilon}^{-1})\|\cdot\|\gamma(T)\|\cdot\|\gamma(C-C_{\varepsilon})\|$
$\displaystyle<$ $\displaystyle
K^{2}\varepsilon\bigl{(}\|\gamma(C^{-1})\|\cdot\|\gamma(C_{\varepsilon}^{-1})\|\cdot\|\gamma(C)\|+\|\gamma(C_{\varepsilon}^{-1})\|\bigr{)}\|\gamma(T)\|.$

∎

######  Lemma 5.

$\alpha(T)\in C^{*}(X)$ for any $T\in C^{*}_{u}(D)$.

###### Proof.

As $T\in C^{*}_{u}(D)$, it can be approximated by finite propagation operators
$T^{N}$, $N\in\mathbb{N}$, with propagation $N$. This means that the matrix of
$T^{N}$ has the property that $t^{N}_{uv}=0$ when $d(u,v)>c$ for some $c>0$).

As $\alpha(T^{N})$ can be approximated by operators of the form
$\alpha_{\varepsilon}(T^{N})$, it suffices to show that
$\alpha_{\varepsilon}(T^{N})\in C^{*}(X)$.

Let $f\in C_{0}(X)$ has compact support $K\subset X$. Then

$\alpha_{\varepsilon}(T^{N})\pi(f)(g)=\sum\nolimits_{u,v\in
D}t^{N}_{uv}\langle\psi_{v}^{\varepsilon},fg\rangle\psi_{u}^{\varepsilon}=\sum\nolimits_{u,v\in
D}t^{N}_{uv}\langle\gamma(C_{\varepsilon})\varphi_{v},fg\rangle\gamma(C_{\varepsilon})\varphi_{u}$
(5)

As $\operatorname{supp}(fg)\subset K$ and as propagation of
$C_{\varepsilon}\leq M_{\varepsilon}$, the set
$\operatorname{supp}(\gamma(C_{\varepsilon})\varphi_{u})$ is intersects only
finitely many simplices. Therefore,
$\langle\gamma(C_{\varepsilon})\varphi_{u},fg\rangle\neq 0$ only for finitely
many $u\in D$. Together with finite propagation of $T^{N}$, this shows that
the sum (5) contains only finite number of non-zero summands, i.e.
$\operatorname{Ran}\alpha_{\varepsilon}(T^{N})\pi(f)$ is finitedimensional.
Similarly, $\operatorname{Ran}\pi(f)\alpha_{\varepsilon}(T^{N})$ is
finitedimensional. Thus $\alpha(T)\pi(f)$ and $\pi(f)\alpha(T)$ are compact.
Approximation of functions in $C_{0}(X)$ by functions $f$ with finite support
proves that $\alpha(T)$ is locally compact.

Similarly one can show that $\alpha(T^{N})$ is of finite propagation. Indeed,
let $f,g\in C_{0}(X)$ are such that the distance between their supports is
greater than $R$. Then

$\pi(f)\alpha_{\varepsilon}(T^{N})\pi(g)(h)=\sum\nolimits_{u,v\in
D}t^{N}_{uv}\langle\gamma(C_{\varepsilon})\varphi_{v},gh\rangle
f\gamma(C_{\varepsilon})\varphi_{u}.$

There exists $L>0$ such that if $d(v,\operatorname{supp}g)>L$ then
$\langle\gamma(C_{\varepsilon})\varphi_{v},gh\rangle=0$, and if
$d(u,\operatorname{supp}f)>L$ then $f\gamma(C_{\varepsilon})\varphi_{u}=0$.
Also $t^{N}_{uv}=0$ when $d(u,v)>c$. Thus, if $R>2L+c$ then
$\pi(f)\alpha_{\varepsilon}(T^{N})\pi(g)=0$. ∎

For $S\in C^{*}(X)$ set $(\beta(S))_{uv}=\langle\psi_{u},S\psi_{v}\rangle$,
$u,v\in D$. Then the operator $\beta(S)\in\mathbb{B}(l^{2}(D))$ can be written
as $\beta(S)(\delta_{v})=\sum_{u\in
D}\langle\psi_{u},S\psi_{v}\rangle\delta_{u}$. It can be also written as
$\beta(S)=U^{*}SU$, where $U:l^{2}(D)\to L^{2}(X)$ is the defined above
isometry. In particular, this implies that $\beta(S)$ is bounded for any
bounded operator $S$. Note that $\beta$ is not a homomorphism.

######  Lemma 6.

Let $S\in C^{*}(X)$. Then $\beta(S)\in C^{*}_{u}(D)$.

###### Proof.

It suffices to show that $\beta(S)\in C^{*}_{u}(D)$ for operators of finite
propagation. For an operator $S$ of finite propagation set
$\tilde{S}=U^{*}F^{*}SFU$, where $F=C_{\varepsilon}C^{-1}$. As
$\|1-F\|<\varepsilon\|C^{-1}\|$, $\beta(S)$ can be approximated by operators
of the form $\tilde{S}$. Let us show that $\tilde{S}$ has finite propagation.
We have

$\tilde{S}_{uv}=\langle\psi_{u},F^{*}SF\psi_{v}\rangle=\langle
F\psi_{u},SF\psi_{v}\rangle=\langle\psi_{u}^{\varepsilon},S\psi_{v}^{\varepsilon}\rangle.$

As $\operatorname{supp}\psi_{u}^{\varepsilon}$ is compactly localized at $u$,
and as $S$ has finite propagation, $\tilde{S}_{uv}=0$ when $d(u,v)$ is
sufficiently great. ∎

Note that $\beta\circ\alpha(S)=pS|_{H}$.

As the constructions of the maps $\alpha$ and $\beta$ work for each $D(n)$, we
have two sequences of maps: $\alpha_{n}:C^{*}_{u}(D(n))\to C^{*}(X)$ and
$\beta_{n}:C^{*}(X)\to C^{*}_{u}(D(n))$. We also write $U_{n}:l^{2}(D(n))\to
L^{2}(X)$ for $U$, $U_{\varepsilon;n}$ for $U_{\varepsilon}$,
$\psi_{u}^{\varepsilon;n}$ for $\psi^{\varepsilon}_{u}$, etc.

##  4\. Continuous field of Roe algebras

Continuous fields of $C^{*}$-algebras (aka bundles of $C^{*}$-algebras or
$C(T)$-$C^{*}$-algebras) were introduced by Fell ([5], see also Dixmier ([3],
Section 10)). Recall that a continuous field of $C^{*}$-algebras over a
locally compact Hausdorff space $T$ is a triple $(T,A,\pi_{t}:A\to A_{t})$,
where $A$ and $A_{t}$, $t\in A$, are $C^{*}$-algebras, the $*$-homomorphisms
$\pi_{t}$ are surjective, the family $\\{\pi_{t}\\}_{t\in T}$ is faithful, and
the map $t\mapsto\|\pi_{t}(a)\|$ is continuous for any $a\in A$.

Set $T=\mathbb{N}\cup\\{\infty\\}$, $A_{n}=C^{*}_{u}(D(n))$,
$A_{\infty}=C^{*}(X)$. Let $F(T)=\prod_{t\in T}A_{t}$ denote the
$C^{*}$-algebra of families $(a_{t})_{t\in T}$ such that $a_{t}\in A_{t}$ for
each $t\in T$, and $\sup_{t\in T}\|a_{t}\|<\infty$. Consider two
$C^{*}$-subalgebras in $F(T)$: let

$J=\\{(a_{t})_{t\in T}\in
F(T):a_{\infty}=\lim_{t\to\infty}\|a_{t}\|=0\\}\cong\oplus_{t\in\mathbb{N}}A_{t}$

and

$B=\\{(\beta_{t}(S))_{t\in T}:S\in C^{*}(X)\\}\subset F(T),$

where $\beta_{\infty}(S)=S$ (as $\beta_{n}$ is a contraction for any
$n\in\mathbb{N}$, $(\beta(S))_{t\in T}\in F(T)$).

Let $A=J+B\subset F(T)$ be the $*$-algebra of the sums of elements of the two
$C^{*}$-subalgebras.

######  Lemma 7.

The $*$-algebra $A$ is norm closed.

###### Proof.

First, let us show that

$\sup_{n\in\mathbb{N}}\|\beta_{n}(S)\|=\|S\|=\lim_{n\to\infty}\|\beta_{n}(S)\|.$
(6)

Consider the projections $p_{n}=U_{n}U_{n}^{*}$ in $L^{2}(X)$, where $U_{n}$
is given by $U_{n}(\delta_{u})=\psi^{n}_{u}$. Note that
$\|U_{n}^{*}SU_{n}\|=\|p_{n}Sp_{n}\|$. Let $f\in C_{0}(X)$ be a Lipschitz
function with compact support $K\subset X$, and let $L$ be the Lipschitz
constant for $f$. Set $g_{n}=\sum_{x\in D(n)}f(x)\varphi_{x}^{n}$ be a
piecewise linear function such that $g_{n}(x)=f(x)$ for any $x\in D$. Then
$|f(x)-g_{n}(x)|\leq L\operatorname{diam}(\Delta^{k}(n))$ for any $x\in X$,
where $\Delta^{k}(n)$ is a simplex of maximal dimension from the $n$-th
subdivision of the initial simplicial structure of $X$, hence
$\|f-g_{n}\|_{L^{2}(X)}\leq L\operatorname{diam}(\Delta^{k}(n))\mu(K)$. As
$\operatorname{diam}(\Delta^{k}(n))$ vanishes as $n\to\infty$,
$\|f-g_{n}\|_{L^{2}(X)}$ vanishes as well. Note that $g_{n}$ lies in the
linear span of the functions $\varphi_{x}^{n}$, $x\in D$, so
$g_{n}=p_{n}g_{n}$. As
$\|f-p_{n}f\|_{L^{2}(X)}\leq\|f-p_{n}g_{n}\|_{L^{2}(X)}$, we have
$\lim_{n\to\infty}f-p_{n}f=0$. As Lipschitz functions with compact support are
dense in $L^{2}(X)$, we conclude that the $*$-strong limit of $p_{n}$ is the
identity operator.

Consider the norm closure $\overline{A}$ of $A$ in $F(T)$. Then $J$ is a
closed ideal in $\overline{A}$. Let
$\\{f_{i}+\beta(S_{i})\\}_{i\in\mathbb{N}}$, $f_{i}\in J$, $S_{i}\in
A_{\infty}$, be a Cauchy sequence in $A$. Passing to the quotient
$C^{*}$-algebra $\overline{A}/J$, the sequence
$\\{f_{i}+\beta(S_{i})+J\\}=\\{\beta(S_{i})+J\\}$ is also a Cauchy sequence,
as the quotient $*$-homomorphisms have norm 1 ([3], Section 1.8). Note that

$\displaystyle\|\beta(S)+J\|=\inf_{f\in
J}\|f+\beta(S)\|\geq\lim_{n\to\infty}\|f(n)+\beta_{n}(S)\|=\lim_{n\to\infty}\|\beta_{n}(S)\|=\|S\|,$

hence $\\{S_{i}\\}_{i\in\mathbb{N}}$ is also a Cauchy sequence. As
$A_{\infty}$ is norm closed, this sequence has a limit in $A_{\infty}$. But
then $\\{f_{i}\\}_{i\in\mathbb{N}}$ is also a Cauchy sequence, and as $J$ is
norm closed, its limit lies in $J$. Thus
$\\{f_{i}+\beta(S_{i})\\}_{i\in\mathbb{N}}$ converges in $A$, hence
$\overline{A}=A$. ∎

Define $\pi_{t}:A\to A_{t}$ by $\pi_{t}(f+\beta(S))=f(t)+\beta_{t}(S)$, $t\in
T$. These maps are well defined as $f_{1}+\beta(S_{1})=f_{2}+\beta(S_{2})$
implies that $f_{1}=f_{2}$ and $S_{1}=S_{2}$.

######  Theorem 8.

The triple $(T,A,\pi_{t}:A\to A_{t})$ is a continuous field of
$C^{*}$-algebras.

###### Proof.

Each $\pi_{t}$ is clearly surjective. If
$\pi_{t}(f_{1}+\beta(S_{1}))=\pi_{t}(f_{2}+\beta(S_{2}))$ for any $t\in T$
then, taking $t=\infty$, we conclude that $S_{1}=S_{2}$. Then we see that
$f_{1}(t)=f_{2}(t)$ for any $t\in\mathbb{N}$, hence $f_{1}=f_{2}$. Finally, we
have to check that the map $t\mapsto\|\pi_{t}(a)\|$ is continuous at
$t=\infty$, which follows from (6). ∎

##  5\. Direct limit of uniform Roe algebras

The main observation of this section is existence of a $*$-homomorphism
$j_{n}:C^{*}_{u}(D(n))\to C^{*}(D(n+1))$ such that the diagram

$\textstyle{C^{*}_{u}(D(n))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{n}}$$\scriptstyle{\alpha_{n}}$$\textstyle{C^{*}_{u}(D(n+1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{n+1}}$$\textstyle{C^{*}(X)}$
(7)

commutes.

There is an obvious inclusion of $C^{*}_{u}(D(n))$ into $C^{*}_{u}(D(n+1))$
induced by the inclusion of $D(n)$ into $D(n+1)$, but this inclusion does not
make the diagram (7) commute, so we cannot pass to the direct limit.

Recall that $H_{n}$ is the closed subspace of $L^{2}(X)$ spanned by the
partition of unity $\\{\varphi^{n}_{u}\\}_{u\in D(n)}$, which is a (non-
orthonormal) basis for $H_{n}$. Our observation is the inclusion $H_{n}\subset
H_{n+1}$. Indeed, it is easy to see that

$\varphi_{u}^{n}=\varphi_{u}^{n+1}+\frac{1}{2}\sum\nolimits_{x\in
D(n+1)\setminus\\{u\\};x\sim u}\varphi_{x}^{n+1},$

where $u\in D(n)$, and $x\sim u$ denotes that $u$ and $x$ lie in the same
simplex $\Delta_{\alpha}(n+1)$. Denote the inclusion $H_{n}\subset H_{n+1}$ by
$I_{n}$, and recall that we have constructed isometric isomorphisms
$U_{n}:l^{2}(D(n))\to H_{n}$. Define $W_{n}:l^{2}(D(n))\to l^{2}(D(n+1))$ by
$W_{n}=U_{n+1}^{*}I_{n}U_{n}$. Then $W_{n}$ is an isometry, different from the
one induced by the inclusion of $D(n)$ into $D(n+1)$. Set
$j_{n}(T)=W_{n}TW_{n}^{*}$. Then
$j_{n}:C^{*}_{u}(D_{n})\to\mathbb{B}(l^{2}(D(n+1)))$ is a $*$-homomorphism.

######  Lemma 9.

$j_{n}(T)\in C^{*}_{u}(D(n+1))$ for any $T\in C^{*}_{u}(D(n))$.

###### Proof.

Recall that besides the maps $U_{n}$, we have constructed maps
$U_{\varepsilon;n}:l^{2}(D_{n})\to H_{n}$ by setting
$U_{\varepsilon;n}(\delta_{u})=\psi_{u}^{\varepsilon,n}$ such that for any
$\varepsilon>0$ there exists $L>0$ such that
$\operatorname{supp}\psi_{u}^{\varepsilon,n}$ lies in the ball of radius $L$
centered at $u\in D(n)$, and $\lim_{\varepsilon\to
0}\|U_{n}-U_{\varepsilon;n}\|=0$. As $\langle
U_{\varepsilon;n}(\delta_{u}),\delta_{v}\rangle=\langle\delta_{u},U_{\varepsilon;n}^{*}(\delta_{v}\rangle$,
the support $\operatorname{supp}U_{\varepsilon;n}(\delta_{u})$ also lies in
the same ball. Set
$W_{\varepsilon;n}=U_{\varepsilon;n+1}^{*}I_{n}U_{\varepsilon;n}$. Then
$\lim_{\varepsilon\to 0}\|W_{\varepsilon;n}-W_{n}\|=0$. As all the three
operators $U_{\varepsilon;n+1}^{*}$, $I_{n}$ and $U_{\varepsilon;n}$ have
finite propagation, so $W_{\varepsilon;n}$ has finite propagation too, and if
$T$ has finite propagation then $W_{\varepsilon;n}TW_{\varepsilon;n}^{*}$ also
has finite propagation, hence lies in $C^{*}_{u}(D(n+1))$. Thus we have
approximated $j_{n}(T)$ by finite propagation operators. ∎

Let $C^{*}_{u}(X)$ be the direct limit $C^{*}$-algebra for the sequence of
$C^{*}$-algebras $C^{*}_{u}(D(n))$ and of $*$-homomorphisms $j_{n}$.

######  Corollary 10.

The sequence of maps $\alpha_{n}:C^{*}(D(n))\to C^{*}(X)$ induces an injective
$*$-homomorphism $\alpha:C^{*}_{u}(X)\to C^{*}(X)$.

###### Proof.

Commutativity of the diagram (7) implies that the direct limit map $\alpha$ is
well defined. As each $\alpha_{n}$ is injective, hence an isometry, so is the
limit map. ∎

Let us describe $\alpha(C^{*}_{u}(X))\subset C^{*}(X)$ in more detail. For
each $u\in D(1)$ choose a closed set $F_{u}\subset X$ with the following
properties:

  * •

there exist $r,R>0$ such that $B_{r}(u)\subset F_{u}\subset B_{R}(u)$ for any
$u\in D(1)$;

  * •

$X=\cup_{u\in D(1)}F_{u}$;

  * •

$\mu(F_{u}\cap F_{v})=0$ for any $u,v\in D(1)$, $u\neq v$.

Then

$L^{2}(X)=\oplus_{u\in D(1)}L^{2}(F_{u}),$ (8)

and it is shown in [10] that if we write an operator $T$ on $L^{2}(X)$ as a
matrix $(T_{uv})_{u,v\in D(1)}$ with respect to the decomposition (8) then
$T\in C^{*}(X)$ iff $T_{uv}:L^{2}(F_{v})\to L^{2}(F_{u})$ is compact for any
$u,v\in D(n)$ and $T$ lies in the closure of the operators of finite
propagation. Consider the subset $R_{m,l}(X)\subset C^{*}(X)$ of operators
$T\in C^{*}(X)$ of propagation $\leq l$ such that the rank of $T_{uv}$ does
not exceed $m$ for any $u,v\in D(1)$. If $T,S\in R_{m,l}(X)$ then $TS\in
R_{mb(l),b(l)}(X)$, where $b(l)$ denotes the maximal number of points in a
ball of radius $l$ in $D(1)$ (recall that $D(1)$ is assumed to have bounded
geometry), hence $\cup_{m,l\in\mathbb{N}}R_{m,l}(X)$ is a $*$-subalgebra in
$C^{*}(X)$. Denote its norm closure by $R(X)$.

######  Lemma 11.

One has $\alpha(C^{*}_{u}(X))\subset R(X)$.

###### Proof.

By definition, $T_{uv}=P_{F_{v}}TP_{F_{u}}$, $u,v\in D(1)$, where $P_{F_{u}}$
denotes the projection onto $L^{2}(F_{u})$. If $T$ lies in the range of
$\alpha_{n}$ then $T=p_{n}Tp_{n}$, where $p_{n}$ is the projection onto
$H_{n}\subset L^{2}(X)$, so $T_{uv}=P_{F_{v}}p_{n}Tp_{n}P_{F_{u}}$, and it
suffices to estimate the rank of $P_{F_{v}}p_{n}$. Let $f\in H_{n}$, then
$f=\sum_{w\in D(n)}f_{w}\varphi^{n}_{w}$, $f_{w}\in\mathbb{C}$. Let $\chi_{F}$
denote the characteristic function of the set $F\subset X$. Then the
projection of $f\in L^{2}(X)$ onto $L^{2}(F_{u})$ equals $\chi_{F_{u}}f$.
Thus, $P_{F_{v}}f=\sum_{w\in D(n)}f_{w}\chi_{F_{v}}\varphi^{n}_{w}$. The
product $\chi_{F_{v}}\varphi^{n}_{w}$ is non-zero only when
$\operatorname{supp}\varphi^{n}_{w}\cap F_{w}\neq\emptyset$, and there exists
$M(n)>0$ such that $\\#\\{w\in D(n):\chi_{F_{v}}\varphi^{n}_{w}\neq 0\\}\leq
M(n)$ for any $v\in D(1)$. Thus, $\operatorname{rk}((\alpha_{n}(T))_{uv})\leq
M(n)$ for any $u,v\in D(1)$. ∎

It would be interesting to find the range of the map $\alpha$ at least for the
case when $X$ is a manifold and the sets $F_{u}$ can be the Voronoi cells
isometric to each other.

##  6\. Remark on $K$-theory

Recall that a continuous field of $C^{*}$-algebras induces a homomorphism in
$K$-theory. Let $S\in C^{*}(X)$, $(\beta_{t}(S))_{t\in T}\in A$, hence
$(\beta_{t}(S)^{2})_{t\in T}\in A$. If $S\in C^{*}(X)$ is a projection, i.e.
satisfies $S=S^{*}=S^{2}$, then
$\lim_{n\to\infty}\|\beta_{n}(S)^{2}-\beta_{n}(S)\|=\|\beta_{\infty}(S)^{2}-\beta_{\infty}(S)\|=\|S^{2}-S\|=0$,
hence $\beta_{n}(S)$ is an almost projection for sufficiently great
$n\in\mathbb{N}$. This also works for projections and for unitaries in matrix
algebras, and equivalent projections give equivalent almost projections, so
this gives a map from $K_{*}(C^{*}(X))$ to $K_{*}(C^{*}_{u}(X))$ (different
projections give almost projections in $C^{*}_{u}(D(n))$ for different $n$,
but all of them can be moved to the direct limit algebra $C^{*}_{u}(X)$ using
the maps $j_{n}$). The inclusion $C^{*}_{u}(X)\subset C^{*}(X)$ induces a map
in $K$-theory in the opposite direction. It is interesting to evaluate the
compositions of these two maps for general spaces. Here we show that non-
trivial elements of $K_{*}(C^{*}(X))$ are mapped to non-trivial elements in
$K_{*}(C^{*}_{u}(D(n)))$ for the simplest non-trivial case $X=\mathbb{R}$.

Recall that the generator of $K_{1}(C^{*}(\mathbb{R}))$ can be given by the
unitary $V$ in the unitalization of $C^{*}(\mathbb{R})$ given by $V=PU+(1-P)$,
where $(Uf)(s)=f(s-1)$ is the shift on $\mathbb{R}$, and $P$ is the projection
onto the closure of the linear span of the characteristic functions
$\chi_{k}=U^{k}(\chi_{[-\frac{1}{2},\frac{1}{2}]})$, $k\in\mathbb{Z}$, where
$\chi_{[a,b]}$ denotes the characteristic function of the segment
$[a,b]\subset\mathbb{R}$ [10]. (In fact, any function in
$L^{2}([-\frac{1}{2},\frac{1}{2}])$ would do instead of $\chi_{0}$.)

######  Theorem 12.

$[\alpha_{n}(\beta_{n}(V))]=[V]$ in $K_{1}(C^{*}(\mathbb{R}))$ for
sufficiently great $n$.

###### Proof.

Note that $D(n)=\frac{1}{2^{n-1}}\mathbb{Z}\subset\mathbb{R}$. Therefore,
$\beta_{n}(U)$ is a shift in $l^{2}(D(n))$,
$\beta_{n}(U)(\delta_{u})=\delta_{u+2^{n-1}}$, $u\in D(n)$. Let us find a
projection in $C^{*}_{u}(D(n))$ close to $\beta_{n}(P)$. Set
$\tilde{\omega}^{n}_{k}=\sum\varphi^{n}_{u}$, where this sum is taken over
those $u\in D(n)$, for which
$\operatorname{supp}\varphi^{n}_{k}\subset[k-\frac{1}{2},k+\frac{1}{2}]$;
$\omega^{n}_{k}=\frac{\tilde{\omega}^{n}_{k}}{\|\tilde{\omega}^{n}_{k}\|}$.
Then $\\{\omega^{n}_{k}\\}_{k\in\mathbb{Z}}$ is an orthonormal system in
$L^{2}(\mathbb{R})$, and $\lim_{n\to\infty}\|\omega^{n}_{k}-\chi_{k}\|=0$. Let
$q^{n}_{k}$ be the projection onto $\omega^{n}_{k}$ in
$L^{2}([k-\frac{1}{2},k+\frac{1}{2}])$,
$Q^{n}(f)=\oplus_{k\in\mathbb{Z}}q^{n}_{k}$. Then

$\|P-Q^{n}\|=\sup_{k\in\mathbb{Z}}\|p_{k}-q^{n}_{k}\|\leq
2\|\omega^{n}_{0}-\chi_{0}\|,$

where $p_{k}$ is the projection onto $\chi_{k}$, vanishes as $n\to\infty$.
Then $\lim_{n\to\infty}\|\beta_{n}(P)-\beta_{n}(Q^{n})\|=0$.

Recall that the bases $\\{\varphi^{n}_{u}\\}$ and $\\{\psi^{n}_{u}\\}$, $u\in
D(n)$, are related by $\psi^{n}_{v}=\sum_{u\in D(n)}c^{n}_{uv}\varphi_{u}$,
where the operator $C_{n}$ with the matrix $(c^{n}_{uv})_{u,v\in D(n)}$ is
invertible and lies in $C^{*}_{u}(\mathbb{Z})$, hence $C^{-1}_{n}$ can be
approximated by operators with band matrices. Let $(d_{uv}^{n})_{u,v\in D(n)}$
be the matrix of $C^{-1}_{n}$. Then we can write $\varphi^{n}_{v}=\sum_{u\in
D(n)}d^{n}_{uv}\psi^{n}_{u}$, hence $\tilde{\omega}^{n}_{k}=\sum_{v}\sum_{u\in
D(n)}d^{n}_{uv}\psi^{n}_{u}$, where the sum over $v$ is taken for
$\operatorname{supp}\varphi^{n}_{v}\subset[k-\frac{1}{2},k+\frac{1}{2}]$.

Then $\beta_{n}(Q^{n})$ is the projection onto the linear span of
$g^{n}_{k}=\sum_{v}\sum_{u\in D(n)}d^{n}_{uv}\delta_{u}\in l^{2}(D(n))$ (with
the same condition on the sum over $v$). Approximating $C^{-1}_{n}$ by
matrices with finite number $N$ of diagonals, we may approximate $g^{n}_{k}$
by $\tilde{g}^{n}_{k}=\sum_{v}\sum_{u\in
D(n);|v-u|\leq\frac{N}{2^{n-1}}}d^{n}_{uv}\delta_{u}$. As we have
$d^{n}_{u+\frac{1}{2^{n-1}},v+\frac{1}{2^{n-1}}}=d^{n}_{uv}$, it is easy to
see that $\lim_{n\to\infty}\|\tilde{g}^{n}_{k}-f^{n}_{k}\|=0$, where
$f^{n}_{k}=\frac{1}{2^{n-1}}\sum_{u\in(k-\frac{1}{2},k+\frac{1}{2})}\delta_{u}$.
Therefore, $\lim_{n\to 0}\|\beta_{n}(Q^{n})-R_{n}\|$, where
$R_{n}=\oplus_{k\in\mathbb{Z}}r^{n}_{k}$, and $r^{n}_{k}$ denotes the
projection onto $f^{n}_{k}$. Now we have
$\lim_{n\to\infty}\|\beta_{n}(P)-R_{n}\|=0$, and
$[\beta_{n}(V)]=[\beta_{n}(U)R_{n}+(1-R_{n})]$ in $K_{1}(C^{*}_{u}(D(n)))$,
which is non-trivial.

As $r^{n}_{k}$ is a rank one projection, it is equivalent to the projection
$s^{n}_{k}$ onto $\delta_{2^{n-1}k}$. Then $\alpha_{n}(R_{n})$ is equivalent
to the projection $\alpha_{n}(S_{n})$ onto the linear span of the functions
$\psi_{2^{n-1}k}$, $k\in\mathbb{N}$. Also
$\alpha_{n}(\beta_{n}(U))\psi_{u}=\psi_{u+2^{n-1}}$, thus
$[\alpha_{n}(\beta_{n}(V))]=[\alpha_{n}(S_{n})W+(1-\alpha_{n}(S_{n}))]$, where
$W\in\mathbb{B}(L^{2}(\mathbb{R}))$ is given by $(Wf)(y)=y-2^{n-1}$, and,
clearly, $[\alpha_{n}(S_{n})W+(1-\alpha_{n}(S_{n}))]$ is equivalent to $V$. ∎

## References

  * [1] J. V. Bellisard. Delone Sets and Material Science: a Program. in Mathematics of Aperiodic Order. Progress in Mathematics, 309, 405–428, Birkhäuser, 2015.
  * [2] A. Connes. Noncommutative Geometry. Academic Press, 1995.
  * [3] J. Dixmier. $C^{*}$-Algebras. North-Holland, 1977.
  * [4] G. A. Elliott, T. Natsume, R. Nest. The Heisenberg group and $K$-theory. $K$-Theory, 7  (1991), 409–428.
  * [5] J. M. G. Fell. The structure of algebras of operator fields. Acta Math. 106 (1961), 233–280.
  * [6] N. Higson, J. Roe. Analytic K-Homology. Oxford Mathematical Monographs, OUP Oxford, 2000.
  * [7] Yu. Kordyukov, V. Manuilov. On topological obstructions to the existence of non-periodic Wannier bases. ArXiv.org: 2304.00125.
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# On the MacWilliams Theorem over Codes and Lattices

Zhiyong Zheng1,2,3, Fengxia Liu∗2,1,3 and Kun Tian∗1,2,3
1Engineering Research Center of Ministry of Education for Financial Computing
and Digital Engineering, Renmin University of China, Beijing, 100872, China
2The Great Bay University, Dongguan, 523830, China
3Henan Academy of Sciences, Zhengzhou, 450046, China
∗Corresponding author email<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract Analogies between codes and lattices have been extensively studied
for the last decades, in this dictionary, the MacWilliams identity is the
finite analog of the Jacobi-Poisson formula of the Theta function. Motivated
by the random theory of lattices, the statistical significance of MacWilliams
theorem is considered, indeed, MacWilliams distribution provides a finite
analog of the classical Gauss distribution. In particular, the MacWilliams
distribution over quotient space of a code is statistical closed with the
uniform distribution.

In the respect of lattices, the analogy of MacWilliams identity associated
with nu-function was conjectured by Solé in 1995. We give an answer to this
problem in positive.

Keywords: MacWilliams Theorem, MacWilliams Distribution, Finite Fourier
Transform, Possion Formula, Smoothing Parameter.

##  1 Introduction

The MacWilliams theorem for linear codes with the Hamming metric [12, 13]
establishes an identity that relates the weight enumerator of a code to the
weight enumerators of its dual code. Various authors have extended this work
in different directions. One direction involves generalizing the weight
enumerators to include more than two variables, such as the Lee and complete
weight enumerators, and extending the concept to codes defined over alphabets
beyond finite fields. For instance, Wan [22] provided a MacWilliams theorem
for codes over Galois rings. Another generalization involves adapting the
notion of weight to consider multiple codewords simultaneously, leading to the
generalized Hamming weights by Wei [23] and MacWilliams-type results for
$m$-tuple support enumerators by Kløve [10], Shiromoto [19], Simonis [20], and
Ray-Chaudhuri and Siap [16]. Britz [4, 5] further generalized some of these
results and provided matroid-theoretic proofs. Additionally, Britz [3]
outlined new and extensive connections between weight enumerators and Tutte
polynomials of matroids.

We first consider the MacWilliams theorem for effective length weight
enumerators of an $m$-tuple of codes $C_{1},C_{2},\cdots,C_{m}$, which need
not be identical. While similar results have been presented by Britz [4],
Kaplan [9], and Shiromoto [19], our focus is on the statistical significance
of the MacWilliams identities. We introduce new concepts for a code, such as
the MacWilliams distribution and the smoothing parameter. In fact, we show
that the MacWilliams distribution is a finite analog of the classical Gaussian
distribution in $\mathbb{R}^{n}$. Specifically, we find that the MacWilliams
distribution closely approximates the uniform distribution over the quotient
space of a code.

Analogies between codes and lattices have been extensively studied over the
past few decades. The classical problem of counting lattice points in
Euclidean spheres involves the use of the Jacobi Theta function. In this
context, the MacWilliams formula can be seen as the finite analog of the
Poisson formula for the Jacobi Theta function. The more recent problem of
counting lattice points in pyramids for the $L^{1}$-norm involves the nu-
function of lattices [2, 21]. Both problems have significant applications in
multidimensional vector quantization [2, 18]. In this paper, we present a new
analog of the MacWilliams identity over lattices with the nu-function, a
problem conjectured by Solé [7, 21]. We provide a positive solution to this
problem.

###  1.1 $m$-Tuple MacWilliams Identity

First, we give the necessary definitions to state the MacWilliams’ original
theorem [13]. Let $F_{q}$ be a finite field of $q$ elements, $n$ a positive
integer, and $C\subset F_{q}^{n}$ a linear code. Let $|C|$ be the number of
codewords of $C$, $<a,b>$ be the usual pairing on $F_{q}^{n}$. The Hamming
weight of any $x\in F_{q}^{n}$, denote by $w(x)$, is the number of nonzero
coordinates of $x$. We denote by $W_{C}(z)$, the Hamming weight enumerator of
$C$ in the indeterminates $z$:

$W_{C}(z)=\sum\limits_{x\in C}z^{w(x)},$

or an equivalent homogeneous form in two indeterminates $z_{1}$ and $z_{2}$

$W_{C}(z_{1},z_{2})=\sum\limits_{x\in C}z_{1}^{w(x)}z_{2}^{n-w(x)}.$

Let $C\subset F_{q}^{n}$ be any linear code and $C^{\bot}$ be its dual code,
MacWilliams [8] (or also see [12]) showed that

$W_{C^{\bot}}(z)=\frac{1}{|C|}(1+(q-1)z)^{n}W_{C}(\frac{1-z}{1+(q-1)z}).$
(1.1)

Let $z=z_{1}/z_{2}$, one has the homogeneous form of MacWilliams identity
immediately (see Theorem 9.32 of [11])

$W_{C^{\bot}}(z_{1},z_{2})=\frac{1}{|C|}W_{C}(z_{2}-z_{1},z_{2}+(q-1)z_{1}).$
(1.2)

Some authors consider more than one code at a time, this leads to the
$m$-tuple MacWilliams identity as follows.

Let $C_{1},C_{2},\cdots,C_{m}\subset F_{q}^{n}$ be $m$ codes, not necessary
being the same. Let $F_{q}^{m\times n}$ be the matrix ring of all $m\times n$
matrices over $F_{q}$. A matrix $x\in F_{q}^{m\times n}$, we denote by
$ew(x)$, the effective length weight of $x$, is the number of nonzero columns
of $x$. Let $\underline{C}=C_{1}\times C_{2}\times\cdots\times C_{m}$,
$\underline{C}^{\bot}=C_{1}^{\bot}\times C_{2}^{\bot}\times\cdots\times
C_{m}^{\bot}$, and $|\underline{C}|=|C_{1}|\times\cdots\times|C_{m}|$. We
regard any an element $x=(x_{1},x_{2},\cdots,x_{m})\in\underline{C}$ as a
matrix of $F_{q}^{m\times n}$ by

$x=\begin{pmatrix}x_{1}\\\ x_{2}\\\ \vdots\\\ x_{m}\end{pmatrix}\in
F_{q}^{m\times n},\quad x_{i}\in C_{i}.$

The effective length weight enumerator of $\underline{C}$ is given by

$W_{\underline{C}}^{(m)}(z)=\sum\limits_{x\in\underline{C}}z^{ew(x)}.$

The following theorem is a generalization of MacWilliams identity for $m$
codes, which is a main result of [9].

Theorem 1 Let $C_{1},C_{2},\cdots,C_{m}$ be any $m$ linear codes over
$F_{q}^{n}$, then we have

$W_{{\underline{C}}^{\bot}}^{(m)}(z)=\frac{1}{|\underline{C}|}(1+(q^{m}-1)z)^{n}W_{\underline{C}}^{(m)}(\frac{1-z}{1+(q^{m}-1)z}).$
(1.3)

This result allows one to compare the effective length of $m$-tuple of vectors
draw from different linear codes, and gives a generalization of an earlier
work of Shiromoto [19] concerning the effective length of $m$-tuple vectors
from the same linear code. Taking $z=z_{1}/z_{2}$ in (1.3), one obtain the
homogeneous form of the $m$-tuple MacWilliams identity, which is the Theorem 6
of [9]. We use the finite Fourier transform over $F_{q}^{m\times n}$ to give
an alternative proof. The proof is not only natural and brief, but also
reflects the statistical significance of the MacWilliams theorem.

###  1.2 MacWilliams Distribution

The use of Gaussian distribution in the study of random lattices is a standard
technique (see, for example, [25]). In [1, 14, 17], a few authors used it to
show that some important cryptographic results. In [1], for example, Gaussian
distribution are used to prove that the certain lattice problems are in coNP.
In [14], Micciancio and Regev introduced the smoothing parameter for a
lattice, which plays a key role in proving of from worst-case to average-case
reductions.

In this paper, we introduce the smoothing parameter for code and show that the
statistical significance of the MacWilliams theorem provides a finite analog
of Gaussian distribution over a code. To state our result, let $X$ and $Y$ be
two discrete type random variables over a finite set $\Gamma$, we define the
statistical distance between $X$ and $Y$ by $\Delta(X,Y)$ (see Definition 2.1,
[14])

$\Delta(X,Y)=\frac{1}{2}\sum\limits_{a\in\Gamma}|\text{Pr}\\{X=a\\}-\text{Pr}\\{Y=a\\}|.$

If the statistical distance $\Delta(X,Y)$ can be arbitrary small, we call $X$
and $Y$ are statistical closed on $\Gamma$.

Let $C\subset F_{q}^{n}$ be a code of length $n$, $z\in(0,1)$ a real number,
we define the MacWilliams distribution as follows

$p(x)=z^{w(x)}/\sum\limits_{x\in C}z^{w(x)},\quad x\in C.$

Since $\sum\limits_{x\in C}p(x)=1$, which corresponds a discrete type random
variable taking value on $C$. Let $F_{q}^{n}/C$ be the quotient space, we
define by $D_{C}$, a discrete random variable over $F_{q}^{n}/C$, its
probability distribution function given by

$g(x)=\sum\limits_{c\in C}z^{w(x+c)}/(1+(q-1)z)^{n},\quad x\in F_{q}^{n}/C.$

We note that if $C$ is a linear code, then

$\sum\limits_{x\in F_{q}^{n}/C}g(x)=1.$

Definition 1.1 Let $\epsilon>0$ be a positive real number, $C\subset
F_{q}^{n}$ be a code and $C^{\bot}$ be its dual code, the smoothing parameter
$\eta_{\epsilon}(C)$ given by

$\eta_{\epsilon}(C)=\min\\{z\big{|}\ 0<z<1,\ \sum\limits_{x\in
C^{\bot}\backslash\\{0\\}}(\frac{1-z}{1+(q-1)z})^{w(x)}<\epsilon\\}.$ (1.4)

One of the main result of this paper is to show the following theorem on
MacWilliams distribution.

Theorem 2 Let $C\subset F_{q}^{n}$ be a linear code, then the random variable
$D_{C}$ over $F_{q}^{n}/C$ is statistical closed with the uniform distribution
over $F_{q}^{n}/C$.

This result shows that the MacWilliams theorem provides a finite analog of the
classical Gaussian distribution in $\mathbb{R}^{n}$.

###  1.3 Theta Function and nu-Function over Lattices

Let $\Lambda\subset\mathbb{R}^{n}$ be a lattice of rank $n$, the Theta
function over $\Lambda$ is defined as the formal power series:

$\theta_{\Lambda}(q)=\sum\limits_{x\in\Lambda}q^{|x|^{2}},$

where $q=e^{\pi iz}$, $|x|^{2}=x_{1}^{2}+\cdots+x_{n}^{2}$ is the common
Euclidean norm. It is easily seen that the coefficient $N_{m}$ of $q^{m^{2}}$
in $\theta_{\Lambda}(q)$ counts the number of lattice points at distance $m$
from the origin in $\mathbb{R}^{n}$. The classical Jacobi Theta function is

$\Theta(\xi|z)=\sum\limits_{m=-\infty}^{+\infty}e^{2mi\xi+\pi izm^{2}}.$

Some special examples of Theta function given by $\theta_{3}(q)=\Theta(0|z)$,
which is the Theta function over lattice $\mathbb{Z}$ and
$\theta_{2}(q)=e^{\pi iz/4}\Theta(\frac{\pi z}{2}|z)$, which is the Theta
function over lattice $\mathbb{Z}$ to co-sets, and it is easy to see that
$\theta_{2}(q^{2})=\theta_{2\mathbb{Z}+1}(q)$.

An analog of MacWilliams theorem over lattices is the following Jacobi-Poisson
formula for Theta function (see [21], or proposition 2.1 of [8])

$\theta_{\Lambda^{*}}(e^{\pi
iz})=\text{det}(\Lambda)(\frac{i}{z})^{\frac{\pi}{2}}\theta_{\Lambda}(e^{-\frac{i\pi}{z}}),$
(1.5)

where $\Lambda^{*}$ is the dual lattice of $\Lambda$, and
$\text{det}(\Lambda)$ is the determinant of $\Lambda$.

To explain why the above Poisson formula is a generalization of MacWill-iams
theorem to lattice, we need some connections between codes and lattices, the
most simple way to associate a lattice with a code is the following
construction [21].

Construction A:

Let $C\subset F_{q}^{n}$ be a binary code coordinatized w.r.t a special basis
of $F_{2}^{n}$. A lattice $A(C)$ is constructed by

$A(C)=\\{(x_{1},x_{2},\cdots,x_{n})\in\mathbb{Z}^{n}\ \big{|}\ \exists c\in C\
\text{such that}\ x\equiv c\ (\text{mod}\ 2)\\}.$ (1.6)

It is known that [21]

$\theta_{A(C)}=W_{C}(\theta_{3}(q^{4}),\theta_{2}(q^{4})),$ (1.7)

where $W_{C}(z_{1},z_{2})$ is the weight enumerator of $C$. Indeed, one could
derive the MacWilliams formula over $C$ from the Jacobi-Poisson formula (1.5)
by using construction A and the relation that
$A(C)^{*}=\frac{1}{2}A(C^{\bot})$ ((3) of [21], or lemma 4.3 below).

Motivated by vector quantizing application [2], some authors [2, 18, 21, 24]
replace the $L^{2}$-norm by $L^{1}$-norm, this leads to hyperbolic
trigonometric function appeared, the nu-function over a lattice $\Lambda$ is
defined by

$\nu_{\Lambda}(z)=\sum\limits_{x\in\Lambda}z^{|x|_{1}}=\sum\limits_{n=0}^{+\infty}z^{n}|\\{x\in\Lambda:|x|_{1}=n\\}|,$

where $|x|_{1}=\sum\limits_{i=1}^{n}|x_{i}|$ is the $L^{1}$-norm. Although the
Poisson formula for nu-function is missing, the same analogy of MacWilliams
identity exists. It was conjectured by [21] that

$2^{\frac{n}{2}}\nu_{\Lambda^{*}}(\text{tanh}^{2}(\frac{\beta}{2}))=\text{det}(\Lambda)(\text{sinh}(2\beta))^{\frac{n}{2}}\nu_{\Lambda}(\text{tanh}(\frac{\alpha}{2})),$
(1.8)

where $\Lambda=A(C)$ associated with a binary linear code $C$, and parameters
$\alpha$ and $\beta$ be connected by the relation
$e^{-2\beta}=\text{tanh}(\alpha)$.

The main purpose of this paper is to show that this conjecture, we give a
positive answer to this problem.

The paper is organized as follows. In section 2, we develop the finite Fourier
transform technique and give a brief proof to Theorem 1. In section 3, we give
a proof to Theorem 2 by using MacWilliams identity. We state and prove Theorem
3 in section 4, which gives an answer to conjecture 1 of Solé [21]. Finally,
we give a conclusion in section 5, a few new consideration are presented.

##  2 Finite Fourier Transform over the Matrix Ring

Let $F_{q}$ be a finite field of $q$ elements, $\psi$ be a non-trivial
additive character of $F_{q}$. Suppose that $f$ is a function defined over
$F_{q}^{n}$, the finite Fourier transform FTf (sometimes called Hadamard
transform) given by

$FTf(x)=\sum\limits_{\xi\in F_{q}^{n}}f(\xi)\psi(<x,\xi>),\quad x\in
F_{q}^{n},$

where $<x,\xi>$ is the usual pairing.

First, we generalize this transform to a matrix ring. Let $F_{q}^{m\times n}$
be the ring consisting of all $m\times n$ matrices over $F_{q}$. Any two
matrices $x,y\in F_{q}^{m\times n}$, then $x^{T}y$ is an $n\times n$ square
matrix. Let Tr$(x^{T}y)$ be the trace of $x^{T}y$, then Tr$(x^{T}y)\in F_{q}$.
We denote by $<x,y>=\text{Tr}(x^{T}y)$, the pairing of $x$ and $y$.

Definition 2.1 Suppose that $f$ is a function defined over $F_{q}^{m\times
n}$, the finite Fourier transform FTf of $f$ is defined by

$FTf(x)=\sum\limits_{\xi\in F_{q}^{m\times n}}f(\xi)\psi(<x,\xi>),\quad x\in
F_{q}^{m\times n}.$

It is easy to see that

$f(x)=\frac{1}{q^{mn}}\sum\limits_{\xi\in F_{q}^{m\times
n}}FTf(\xi)\psi(-<x,\xi>),\quad x\in F_{q}^{m\times n}.$

Let $C_{1},C_{2},\cdots,C_{m}$ be $m$ codes over $F_{q}$ of length $n$,
$\underline{C}=C_{1}\times\cdots\times C_{m}$, we regard $\underline{C}$ as a
subset of $F_{q}^{m\times n}$ according to $x=(x_{1},\cdots,x_{m})^{T}$, where
$x_{i}\in C_{i}$. Let $\chi_{\underline{C}}$ be the characteristic function of
$\underline{C}$, that means $\chi_{\underline{C}}(x)=1$, if
$x\in\underline{C}$, and $\chi_{\underline{C}}(x)=0$, if
$x\notin\underline{C}$.

Lemma 2.1 If $C_{i}$ ($1\leqslant i\leqslant m$) is the linear code of
$F_{q}^{n}$, then we have

$\chi_{\underline{C}^{\bot}}(x)=\frac{1}{|\underline{C}|}\text{FT}\chi_{\underline{C}}(x),$

where $\underline{C}^{\bot}=C_{1}^{\bot}\times C_{2}^{\bot}\times\cdots\times
C_{m}^{\bot}$.

Proof: By the definition of FT$\chi_{\underline{C}}$, we see that for any
$x\in F_{q}^{m\times n}$,

$\text{FT}\chi_{\underline{C}}(x)=\sum\limits_{\xi\in F_{q}^{m\times
n}}\chi_{\underline{C}}(\xi)\psi(<x,\xi>)=\sum\limits_{\xi\in\underline{C}}\psi(<x,\xi>).$

If we write

$x=\begin{pmatrix}x_{1}\\\ \vdots\\\ x_{m}\end{pmatrix},\ x_{i}\in F_{q}^{n},\
\text{and}\ \xi=\begin{pmatrix}\xi_{1}\\\ \vdots\\\ \xi_{m}\end{pmatrix},\
\xi_{i}\in C_{i},$

then

$<x,\xi>=\text{Tr}(x^{T}\xi)=\sum\limits_{i=1}^{m}<x_{i},\xi_{i}>.$ (2.1)

It follows that

$\text{FT}\chi_{\underline{C}}(x)=\mathop{\prod}\limits_{i=1}^{m}\sum\limits_{\xi_{i}\in
C_{i}}\psi(<x_{i},\xi_{i}>).$

It is known that (see, for example, lemma 9.31 of [11])

$\sum\limits_{\xi_{i}\in
C_{i}}\psi(<x_{i},\xi_{i}>)=\left\\{\begin{array}[]{cc}|C_{i}|,&\text{if}\
x_{i}\in C_{i}^{\bot}.\\\ 0,&\text{otherwise.}\end{array}\right.$

We have
FT$\chi_{\underline{C}}(x)=|\underline{C}|\chi_{\underline{C}^{\bot}}(x)$.
$\Box$

Using the above observation, we have the following more general Poisson
formula over matrix ring.

Lemma 2.2 Let $\underline{C}\subset F_{q}^{m\times n}$ be the block linear
codes over $F_{q}$, and $f(x)$ be arbitrary function over $F_{q}^{m\times n}$,
then

$\sum\limits_{x\in\underline{C}^{\bot}}f(x)=\frac{1}{|\underline{C}|}\sum\limits_{x\in\underline{C}}\text{FT}f(x).$

Proof:

$\sum\limits_{x\in\underline{C}^{\bot}}f(x)=\sum\limits_{x\in F_{q}^{m\times
n}}\chi_{\underline{C}^{\bot}}(x)f(x)$
$\qquad\qquad\qquad\quad=\frac{1}{|\underline{C}|}\sum\limits_{x\in
F_{q}^{m\times n}}f(x)\text{FT}\chi_{\underline{C}}(x)$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\frac{1}{|\underline{C}|}\sum\limits_{\xi\in
F_{q}^{m\times n}}\chi_{\underline{C}}(\xi)\sum\limits_{x\in F_{q}^{m\times
n}}f(x)\psi(<x,\xi>)$
$\qquad\quad=\frac{1}{|\underline{C}|}\sum\limits_{\xi\in\underline{C}}\text{FT}f(\xi).$

$\Box$

Now, we give a brief proof to Theorem 1.

Proof of Theorem 1:

Let $f(x)=z^{ew(x)}$, where $x\in F_{q}^{m\times n}$, and $ew(x)$ is the
effective length weight function given by section 1. To prove Theorem 1, by
the Poisson formula, we only show that

$\text{FT}z^{ew(x)}=(1+(q^{m}-1)z)^{n}(\frac{1-z}{1+(q^{m}-1)z})^{ew(x)}.$
(2.2)

We write $x=(x_{1},x_{2},\cdots,x_{n})\in F_{q}^{m\times n}$, $x_{i}\in
F_{q}^{m}$, and $\xi=(\xi_{1},\xi_{2},\cdots,\xi_{n})$ $\in F_{q}^{m\times
n}$, $\xi_{i}\in F_{q}^{m}$, then (comparing with (2.1))

$<x,\xi>=\text{Tr}(x^{T}\xi)=\sum\limits_{i=1}^{n}<\xi_{i},x_{i}>.$ (2.3)

It follows that

$\text{FT}f(x)=\sum\limits_{\xi\in F_{q}^{m\times n}}f(\xi)\psi(<x,\xi>)$
$\qquad\qquad\ =\sum\limits_{\xi\in F_{q}^{m\times
n}}z^{ew(\xi)}\psi(<x,\xi>)$ $\qquad\qquad\qquad\qquad\ \ \
=\sum\limits_{\xi\in F_{q}^{m\times
n}}z^{ew(\xi_{1})+\cdots+ew(\xi_{n})}\psi(<x,\xi>)$

where $ew(\xi_{i})=1$, if $\xi_{i}\neq 0$, and $ew(\xi_{i})=0$, if $\xi_{i}=0$
is a zero vector. Therefore, we have

$\text{FT}f(x)=\mathop{\prod}\limits_{i=1}^{n}(\sum\limits_{\xi_{i}\in
F_{q}^{m}}z^{ew(\xi_{i})}\psi(<x_{i},\xi_{i}>))$
$\qquad\qquad\quad=\mathop{\prod}\limits_{i=1}^{n}(1+z\sum\limits_{\xi_{i}\in
F_{q}^{m}\backslash\\{0\\}}\psi(<x_{i},\xi_{i}>)).$

It is known that

$\sum\limits_{\xi_{i}\in
F_{q}^{m}\backslash\\{0\\}}\psi(<x_{i},\xi_{i}>)=\left\\{\begin{array}[]{cc}q^{m}-1,&\text{if}\
x_{i}=0.\\\ -1,&\text{if}\ x_{i}\neq 0.\end{array}\right.$

It follows that

$\text{FT}z^{ew(x)}=\mathop{\prod}\limits_{i=1}^{n}(1+z\sum\limits_{\xi_{i}\in
F_{q}^{m}\backslash\\{0\\}}\psi(<x_{i},\xi_{i}>))\qquad$ $\qquad\qquad\quad\
=\mathop{\prod}\limits_{i=1}^{n}((1-z)^{ew(x_{i})}(1+(q^{m}-1)z)^{1-ew(x_{i})})$
$\qquad\qquad\ =(1+(q^{m}-1)z)^{n}(\frac{1-z}{1+(q^{m}-1)z})^{ew(x)}.$ (2.4)

We complete the proof of Theorem 1.
$\Box$

In the proof of Theorem 1, it is important that the function $z^{ew(x)}$ is
almost fixed under the finite Fourier transform over $F_{q}^{m\times n}$, thus
this function provides a finite analog of Gaussian measure, and we obtain a
finite analog of Gaussian distribution by using the fixed point. The
statistical significance of MacWilliams theorem is revealed by this
observation.

##  3 Proof of Theorem 2

Let $C\subset F_{q}^{n}$ be a linear code of length $n$, $C^{\bot}$ be its
dual code, $0<z<1$ be a real number parameter. The smoothing parameter
$\eta_{\epsilon}(C)$ is defined as (see Definition 1.1)

$\eta_{\epsilon}(C)=\min\\{z\ \big{|}\ 0<z<1,\sum\limits_{x\in
C^{\bot}\backslash\\{0\\}}(\frac{1-z}{1+(q-1)z})^{w(x)}<\epsilon\\}.$ (3.1)

If we take $C=0$, a zero code in MacWilliams identity, it is easy to see that

$\sum\limits_{x\in F_{q}^{n}}z^{w(x)}=(1+(q-1)z)^{n}.$

The following probability distribution function corresponds a discrete random
variable $D_{C}$ over the quotient space $F_{q}^{n}/C$

$g(x)=\sum\limits_{c\in C}z^{w(x+c)}/(1+(q-1)z)^{n},\quad x\in F_{q}^{n}/C.$

To show that $D_{C}$ is statistical closed to the uniform distribution
$U(F_{q}^{n}/C)$, we only prove the following proposition.

Proposition 3.1 Let $C\subset F_{q}^{n}$ be a linear code, then for arbitrary
$\epsilon>0$ and $\eta_{\epsilon}(C)\leqslant z<1$, the statistical distance
$D_{C}$ and the uniform distribution $U(F_{q}^{n}/C)$ satisfies the following
inequality:

$\Delta(D_{C},U(F_{q}^{n}/C))\leqslant\frac{1}{2}\epsilon.$

Proof: Let $f(x)$ be any a function over $F_{q}^{n}$, by lemma 2.2 we have the
Poisson formula over code $C$,

$\sum\limits_{x\in C^{\bot}}f(x)=\frac{1}{|C|}\sum\limits_{x\in
C}\text{FT}f(x).$ (3.2)

If $x\in F_{q}^{n}/C$ and $z$ are given, let

$f(c)=z^{w(x+c)}/(1+(q-1)z)^{n},\quad\forall c\in C.$

The probability distribution function $g(x)$ of $D_{C}$ is

$g(x)=\sum\limits_{c\in C}f(c),\quad\forall x\in F_{q}^{n}/C.$

First, we calculate the Fourier transform FT$f(c)$. By definition,

$\text{FT}f(c)=\sum\limits_{\xi\in
F_{q}^{n}}f(\xi)\psi(<\xi,c>)\qquad\qquad\qquad\qquad$
$\qquad\qquad=\frac{1}{(1+(q-1)z)^{n}}\sum\limits_{\xi\in
F_{q}^{n}}z^{w(x+\xi)}\psi(<\xi,c>).$

Let $x+\xi=\xi_{1}$. By (2.4) we have ($m=1$ in (2.4))

$\text{FT}f(c)=\frac{\psi(-<x,c>)}{(1+(q-1)z)^{n}}\sum\limits_{\xi_{1}\in
F_{q}^{n}}z^{w(\xi_{1})}\psi(<\xi_{1},c>)$ $\ \
=(\frac{1-z}{1+(q-1)z})^{w(c)}\psi(-<x,c>).$

According to the Poisson formula (3.2), it follows that

$g(x)=\sum\limits_{c\in C}f(c)=\frac{1}{|C^{\bot}|}\sum\limits_{c\in
C^{\bot}}\text{FT}f(c)\qquad\quad$ $\qquad\qquad\quad\
=\frac{1}{|C^{\bot}|}\sum\limits_{c\in
C^{\bot}}(\frac{1-z}{1+(q-1)z})^{w(c)}\psi(-<x,c>).$

Therefore, the statistical distance $\Delta(D_{C},U(F_{q}^{n}/C))$ may be
calculated by

$\Delta(D_{C},U(F_{q}^{n}/C))=\frac{1}{2}\sum\limits_{x\in
F_{q}^{n}/C}\Big{|}g(x)-\frac{1}{|C^{\bot}|}\Big{|}$
$=\frac{1}{2|C^{\bot}|}\sum\limits_{x\in F_{q}^{n}/C}\Big{|}\sum\limits_{c\in
C^{\bot}}(\frac{1-z}{1+(q-1)z})^{w(c)}\psi(-<x,c>)-1\Big{|}$
$=\frac{1}{2|C^{\bot}|}\sum\limits_{x\in F_{q}^{n}/C}\Big{|}\sum\limits_{c\in
C^{\bot}\backslash\\{0\\}}(\frac{1-z}{1+(q-1)z})^{w(c)}\psi(-<x,c>)\Big{|}$
$\leqslant\frac{1}{2}\Big{|}\sum\limits_{c\in
C^{\bot}\backslash\\{0\\}}(\frac{1-z}{1+(q-1)z})^{w(c)}\Big{|}.\qquad\qquad\qquad\qquad\qquad\
\ \ $

If $\eta_{\epsilon}(C)\leqslant z<1$, we have

$\Big{|}\sum\limits_{c\in
C^{\bot}\backslash\\{0\\}}(\frac{1-z}{1+(q-1)z})^{w(c)}\Big{|}\leqslant\epsilon.$

It follows that

$\Delta(D_{C},U(F_{q}^{n}/C))\leqslant\frac{1}{2}\epsilon.$

We complete the proof of proposition 3.1.
$\Box$

##  4 The nu-Function over Lattices

Let $\Lambda\subset\mathbb{R}^{n}$ be a lattice of rank $n$, the nu-function
over $\Lambda$ is defined by

$\nu_{\Lambda}(z)=\sum\limits_{x\in\Lambda}z^{|x|_{1}},$

where $|x|_{1}=|x_{1}|+|x_{2}|+\cdots+|x_{n}|$ is the $L^{1}$-norm. A function
$f$ defined over $\mathbb{R}^{n}$, the Fourier transform given by

$\text{FT}f(x)=\int_{\mathbb{R}^{n}}f(\xi)e^{-2\pi
i<x,\xi>}\mathrm{d}\xi,\quad x\in\mathbb{R}^{n},$

where $<x,\xi>=\sum\limits_{i=1}^{n}x_{i}\xi_{i}$ is the usual pairing of $x$
and $\xi$. It is easy to see that the Gauss function $e^{-\pi|x|^{2}}$ is a
fixed point under the Fourier transform, or slightly more general, with a
parameter $s$, one has

$\text{FT}e^{-\pi|x/s|^{2}}=s^{n}e^{-\pi|sx|^{2}}.$

Indeed, the MacWilliams weight enumerator $z^{w(x)}$ is a finite analog of
Gauss function. The following Poisson formula over lattices is classical (see
Theorem 2.3 of [8], or lemma 1.9 of [25]).

Lemma 4.1 Let $\Lambda\subset\mathbb{R}^{n}$ be a lattice of rank $n$,
$f:\mathbb{R}^{n}\rightarrow\mathbb{C}$ be a function which satisfies the
following conditions (V1), (V2), (V3):

(V1) $\int_{\mathbb{R}^{n}}|f(x)|\mathrm{d}x<\infty$.

(V2) The series $\sum\limits_{x\in\Lambda}|f(x+u)|$ converges uniformly for
all $u$ belonging to a compact subset of $\mathbb{R}^{n}$.

(V3) The series $\sum\limits_{x\in\Lambda^{*}}\text{FT}f(x)$ is absolutely
convergent.
Then we have

$\sum\limits_{x\in\Lambda}f(x)=\frac{1}{\text{det}(\Lambda)}\sum\limits_{x\in\Lambda^{*}}\text{FT}f(x).$

Taking $f(x)=e^{\pi iz|x|^{2}}$ in lemma 4.1, we have the Jacobi-Poisson
formula (see (1.5)) immediately. If one replace the $L^{2}$-norm by
$L^{1}$-norm, and consider the nu-function over $\Lambda$, since $z^{|x|_{1}}$
is not a fixed point under the Fourier transform, the Poisson formula for nu-
function over lattice $\Lambda$ is missing.

For some special lattices, for example $A(C)$, the lattices associated with
the binary code $C$ by using construction A, the analogy of MacWilliams
theorem for nu-function still exists. We state our main result as follows.

Theorem 3 Let $\Lambda=A(C)$, $\alpha$ and $\beta$ be the parameters such that
$e^{-2\beta}=\tanh(\alpha)$, then

$2^{\frac{n}{2}}\nu_{\Lambda^{*}}(\tanh^{2}(\frac{\beta}{2}))=\text{det}(\Lambda)(\sinh(2\beta))^{\frac{n}{2}}\nu_{\Lambda}(\tanh(\frac{\alpha}{2})),$

where $\tanh$ and $\sinh$ are hyperbolic trigonometric functions.

This result is an open problem appeared in Solé [21] as the Conjecture 1, but
in his original statement the relation $e^{-2\beta}=\tanh(\alpha)$ is missed
by $e^{-\beta}=\tanh(\alpha)$. Of course, that is a print error, because he
already noted the symmetric relation $e^{-2\alpha}=\tanh(\beta)$. To show that
Theorem 3, we first prove a few auxiliary lemmas.

Lemma 4.2 Let $L\subset\mathbb{R}^{n}$ be any a lattice, $z$ is a given
parameter, then $\nu_{2L}(z)=\nu_{L}(z^{2})$.

Proof: By the definition of nu-function, we have

$\nu_{2L}(z)=\sum\limits_{x\in 2L}z^{|x|_{1}}=\sum\limits_{y\in
L}z^{|2y|_{1}}=\sum\limits_{y\in L}(z^{2})^{|y|_{1}}=\nu_{L}(z^{2}).$

$\Box$

Lemma 4.3 Let $A(C)$ be a lattice associated with the binary code $C$, then
the dual lattice of $A(C)$ given by

$A(C)^{*}=\frac{1}{2}A(C^{\bot}),$

where $C^{\bot}$ is the dual code of $C$.

Proof: We first prove that $A(C)^{*}\subset\frac{1}{2}A(C^{\bot})$. For any
$\alpha\in A(C)^{*}$, we note that $x\in A(C)$ whence $x\in C$, it follows
that $<\alpha,x>\in\mathbb{Z}$ for all $x\in C$, and

$<2\alpha,x>\equiv 0\ (\text{mod}\ 2),\quad\forall x\in C.$

This means that $2\alpha\ \text{mod}\ 2\in C^{\bot}$, and $2\alpha\in
A(C^{\bot})$, which implies that $\alpha\in\frac{1}{2}A(C^{\bot})$.

To show that $\frac{1}{2}A(C^{\bot})\subset A(C)^{*}$, for any
$\beta\in\frac{1}{2}A(C^{\bot})$, or $2\beta\in A(C^{\bot})$, we see that
$\beta\in A(C)^{*}$. Let $x\in C$, we have $<2\beta\ \text{mod}\ 2,x>=0$,
which implies that $<2\beta,x>\equiv 0\ (\text{mod}\ 2)$. Thus we have
$<\beta,x>\in\mathbb{Z}$ for all $x\in C$. Let $y\in A(C)$, denote $y\
\text{mod}\ 2=x_{0}\in C$, then $<\beta,y>\in\mathbb{Z}$ by
$<\beta,x_{0}>\in\mathbb{Z}$, this leads to $\beta\in A(C)^{*}$, and
$\frac{1}{2}A(C^{\bot})\subset A(C)^{*}$. We have lemma 4.3.
$\Box$

Lemma 4.4 Let $C\subset F_{2}^{n}$ be a binary code of length $n$, $\alpha$ be
any a parameter, then we have

$\nu_{A(C)}(\tanh\frac{\alpha}{2})=W_{C}(\cosh\alpha,\sinh\alpha),$

where $W_{C}(z_{1},z_{2})$ is the weight enumerator in the homogeneous form.

Proof: For any $x\in C$, we define a set $A(x)$ by

$A(x)=\\{y\in\mathbb{Z}^{n}\ |\ y\equiv x\ (\text{mod}\ 2)\\}.$

It is easy to see that $A(x_{1})\cap A(x_{2})=\emptyset$, if $x_{1}\neq
x_{2}$, so we have

$A(C)=\mathop{\cup}\limits_{x\in C}A(x),\ \text{and}\
\nu_{A(C)}(z)=\sum\limits_{x\in C}\nu_{A(x)}(z).$

Let $x=(x_{1},x_{2},\cdots,x_{n})$, then
$A(x)=(x_{1}+2\mathbb{Z})\times(x_{2}+2\mathbb{Z})\times\cdots\times(x_{n}+2\mathbb{Z})$.
It is not difficult to see

$\nu_{A(x)}(z)=\sum\limits_{y\in A(x)}z^{|y|_{1}}=(\sum\limits_{y^{\prime}\in
2\mathbb{Z}}z^{|y^{\prime}|})^{n-w(x)}(\sum\limits_{y^{\prime\prime}\in
2\mathbb{Z}+1}z^{|y^{\prime\prime}|})^{w(x)}$
$=(\nu_{2\mathbb{Z}}(z))^{n-w(x)}(\nu_{2\mathbb{Z}+1}(z))^{w(x)}.\qquad\qquad\
$

It follows that

$\nu_{A(c)}(z)=\sum\limits_{x\in C}\nu_{A(x)}(z)=\sum\limits_{x\in
C}(\nu_{2\mathbb{Z}}(z))^{n-w(x)}(\nu_{2\mathbb{Z}+1}(z))^{w(x)}$
$=W_{C}(\nu_{2\mathbb{Z}}(z),\nu_{2\mathbb{Z}+1}(z)).\qquad\qquad\qquad\quad\
$

If the parameter $z$ satisfies $|z|<1$, one has

$\nu_{2\mathbb{Z}}(z)=\frac{1+z^{2}}{1-z^{2}},\ \text{and}\
\nu_{2\mathbb{Z}+1}(z)=\frac{2z}{1-z^{2}}.$

By the above calculate, we obtain

$\nu_{A(C)}(z)=W_{C}(\frac{1+z^{2}}{1-z^{2}},\frac{2z}{1-z^{2}}).$

Let $\alpha$ be a parameter such that $z=\tanh\frac{\alpha}{2}$, it is easily
seen that

$\frac{1+z^{2}}{1-z^{2}}=\cosh\alpha,\ \text{and}\
\frac{2z}{1-z^{2}}=\sinh\alpha.$

We have

$\nu_{A(C)}(\tanh\frac{\alpha}{2})=W_{C}(\cosh\alpha,\sinh\alpha).$

This is the proof of lemma 4.4.
$\Box$

Lemma 4.5 For any binary code $C\subset F_{2}^{n}$, we have

$|C|\text{det}(A(C))=2^{n},$

where $A(C)$ is the lattice associated with $C$, and $\text{det}(A(C))$ is the
determinant of $A(C)$.

Proof: Considering the additive group homomorphism
$\sigma:\mathbb{Z}^{n}\rightarrow F_{2}^{n}$ given by

$(x_{1},x_{2},\cdots,x_{n})\in\mathbb{Z}^{n}\xrightarrow[]{\ \ \sigma\ \
}(x_{1}\ \text{mod}\ 2,x_{2}\ \text{mod}\ 2,\cdots,x_{n}\ \text{mod}\ 2)\in
F_{2}^{n}.$

Obviously, $\sigma^{-1}(C)=A(C)$, one has the following quotient group
isomorphism

$\mathbb{Z}^{n}/A(C)\cong F_{2}^{n}/C.$

It is known that

$\text{det}(A(C))=|\mathbb{Z}^{n}/A(C)|.$

Thus, we have

$|C|\text{det}(A(C))=|F_{2}^{n}|=2^{n}.$

$\Box$

Lemma 4.6 For any binary code $C\subset F_{2}^{n}$, and two parameters
$\alpha$ and $\beta$ satisfying $e^{-2\beta}=\tanh\alpha$, then we have

$\text{det}(A(C))(\frac{\sinh
2\beta}{2})^{\frac{n}{2}}(\frac{\cosh\alpha}{\cosh\beta})^{n}=\frac{1}{|C|}(1+\tanh\beta)^{n}.$

where $\text{det}(A(C))$ is the determinant of $A(C)$.

Proof: Based on the equality $e^{-2\beta}=\tanh\alpha$ we can get
$e^{-2\alpha}=\tanh\beta$ symmetrically, then

$\cosh^{2}\alpha=\frac{(e^{\alpha}+e^{-\alpha})^{2}}{4}=\frac{e^{2\alpha}+e^{-2\alpha}+2}{4}=(\frac{\cosh\beta}{\sinh\beta}+\frac{\sinh\beta}{\cosh\beta}+2)/4$
$=\frac{(\cosh\beta+\sinh\beta)^{2}}{4\sinh\beta\cosh\beta}=\frac{(\cosh\beta+\sinh\beta)^{2}}{2\sinh
2\beta}.\qquad\qquad$

So we have

$2\sinh 2\beta\cosh^{2}\alpha=(\cosh\beta+\sinh\beta)^{2}$

and

$2\sinh
2\beta(\frac{\cosh\alpha}{\cosh\beta})^{2}=(\frac{\cosh\beta+\sinh\beta}{\cosh\beta})^{2}=(1+\tanh\beta)^{2}.$

Take the $\frac{n}{2}$ power of both sides in the above equality, we get

$2^{\frac{n}{2}}(\sinh
2\beta)^{\frac{n}{2}}(\frac{\cosh\alpha}{\cosh\beta})^{n}=(1+\tanh\beta)^{n}.$

By lemma 4.5, $\text{det}(A(C))|C|=2^{n}$, therefore,

$\text{det}(A(C))|C|2^{-\frac{n}{2}}(\sinh
2\beta)^{\frac{n}{2}}(\frac{\cosh\alpha}{\cosh\beta})^{n}=(1+\tanh\beta)^{n},$

which is equivalent to

$\text{det}(A(C))(\frac{\sinh
2\beta}{2})^{\frac{n}{2}}(\frac{\cosh\alpha}{\cosh\beta})^{n}=\frac{1}{|C|}(1+\tanh\beta)^{n}.$

Then we finish the proof of this lemma.
$\Box$

Now, we give a proof of Theorem 3.

Proof of Theorem 3:

Let $C\subset F_{2}^{n}$ be arbitrary a binary code of length $n$, $A(C)$ be
the lattice associated with $C$. By lemma 4.2, we have

$\nu_{A(C)^{*}}(\tanh^{2}\frac{\beta}{2})=\nu_{2A(C)^{*}}(\tanh\frac{\beta}{2}).$
(4.1)

By lemma 4.3, we have

$\nu_{2A(C)^{*}}(\tanh\frac{\beta}{2})=\nu_{A(C^{\bot})}(\tanh\frac{\beta}{2})$
(4.2)

and by lemma 4.4

$\nu_{A(C^{\bot})}(\tanh\frac{\beta}{2})=W_{C^{\bot}}(\cosh\beta,\sinh\beta).$
(4.3)

Combining with (4.1), (4.2) and (4.3), we have

$\nu_{A(C)^{*}}(\tanh^{2}\frac{\beta}{2})=W_{C^{\bot}}(\cosh\beta,\sinh\beta).$
(4.4)

Similarly, in (4.3) we replace $C^{\bot}$ by $C$, which gives

$\nu_{A(C)}(\tanh\frac{\beta}{2})=W_{C}(\cosh\beta,\sinh\beta).$

Therefore, to prove Theorem 3, it is sufficient to prove

$W_{C^{\bot}}(\cosh\beta,\sinh\beta)=\text{det}(A(C))(\frac{\sinh
2\beta}{2})^{\frac{n}{2}}W_{C}(\cosh\alpha,\sinh\alpha).$ (4.5)

First, we note that

$W_{C}(X,Y)=\sum\limits_{c\in C}X^{n-w(c)}Y^{w(c)}=X^{n}\sum\limits_{c\in
C}(\frac{Y}{X})^{w(c)},$

thus,

$W_{C^{\bot}}(\cosh\beta,\sinh\beta)=(\cosh\beta)^{n}\sum\limits_{c\in
C^{\bot}}(\tanh\beta)^{w(c)}$

and

$W_{C}(\cosh\alpha,\sinh\alpha)=(\cosh\alpha)^{n}\sum\limits_{c\in
C}(\tanh\alpha)^{w(c)}.$

We only prove the following identity

$(\cosh\beta)^{n}\sum\limits_{c\in
C^{\bot}}(\tanh\beta)^{w(c)}=\text{det}(A(C))(\frac{\sinh
2\beta}{2})^{\frac{n}{2}}(\cosh\alpha)^{n}\sum\limits_{c\in
C}(\tanh\alpha)^{w(c)},$

or equivalently,

$\sum\limits_{c\in C^{\bot}}(\tanh\beta)^{w(c)}=\text{det}(A(C))(\frac{\sinh
2\beta}{2})^{\frac{n}{2}}(\frac{\cosh\alpha}{\cosh\beta})^{n}\sum\limits_{c\in
C}(\tanh\alpha)^{w(c)}.$ (4.6)

By the MacWilliams identity of $C$ ($q=2$, see (1.1))

$\sum\limits_{c\in C^{\bot}}z^{w(c)}=\frac{1}{|C|}(1+z)^{n}\sum\limits_{c\in
C}(\frac{1-z}{1+z})^{w(c)}.$

Let $z=\tanh\beta$, note that

$\frac{1-z}{1+z}=e^{-2\beta}=\tanh\alpha,$

we thus have

$\sum\limits_{c\in
C^{\bot}}(\tanh\beta)^{w(c)}=\frac{1}{|C|}(1+\tanh\beta)^{n}\sum\limits_{c\in
C}(\tanh\alpha)^{w(c)}.$ (4.7)

By lemma 4.6, we have the following identity

$\text{det}(A(C))(\frac{\sinh
2\beta}{2})^{\frac{n}{2}}(\frac{\cosh\alpha}{\cosh\beta})^{n}=\frac{1}{|C|}(1+\tanh\beta)^{n}.$
(4.8)

Comparing the right hand sides of (4.6) and (4.7), we have (4.6) by (4.8), and
thus we have (4.5). This is the proof of Theorem 3.
$\Box$

##  5 Conclusion

We show that in this paper, the statistical significance of MacWilliams
theorem is to provide a real instance of the classical Gauss distribution for
a code. If $C$ is a binary code, for example, $A(C)$ is the lattice associated
with $C$, then the discrete Gauss measure over $A(C)$ is corresponding with
the MacWilliams distribution over $C$. The further question is to give a
precise lower bound for the smoothing parameter $\eta_{\epsilon}(C)$ of the
arbitrary codes. Now we have a rough lower bound of it for linear codes as
$\eta_{\epsilon}(C)\geqslant[(\frac{q^{n-k}}{1+\epsilon})^{\frac{1}{n}}-1]/(q-1)$.
We will discuss this problem in the following works.

To generalize the results given by Theorem 3, it is natural to establish the
MacWilliams type formula for the nu-function over arbitrary lattices. As we
already pointed out that the Poisson formula for nu-function is missing, one
of alternative method is to generalize the construction A, or other
constructions for a lattice, so that we may transform this problem to a code
over $F_{q}$, and make use of the MacWilliams identity. However, we don’t have
any new results in this direction.

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