Nonexistence of perfect $2$-error-correcting Lee codes in certain dimensions

TL;DR

This paper proves the nonexistence of certain perfect 2-error-correcting Lee codes in specific dimensions, providing evidence supporting the Golomb--Welch conjecture that such codes do not exist for higher dimensions.

Contribution

It establishes the nonexistence of perfect 2-error-correcting Lee codes for an infinite class of dimensions, advancing understanding of the Golomb--Welch conjecture.

Findings

Proves nonexistence of perfect 2-error-correcting Lee codes in certain dimensions

Supports the Golomb--Welch conjecture for higher dimensions

Provides a new class of dimensions where perfect Lee codes cannot exist

Abstract

The Golomb--Welch conjecture states that there are no perfect $e$-error-correcting codes in $\mathbb{Z}^n$ for $n \ge 3$ and $e \ge 2$. In this note, we prove the nonexistence of perfect $2$-error-correcting codes for a certain class of $n$, which is expected to be infinite. This result further substantiates the Golomb--Welch conjecture.

Tables

Table 1. Table 1. The number of n 𝑛 n to which Theorem  1.4 can be applied

$x$	# of $n\le x$ with $2n^2+2n+1$ prime	# of $n\le x$ to which Theorem 1.4 can be applied
${10}^1$	$6$	$4$
${10}^2$	$36$	$34$
${10}^3$	$225$	$222$
${10}^4$	$1645$	$1642$
${10}^5$	$12706$	$12702$

Equations

$$d(\mathbf{x},\mathbf{y})=\sum _{i=1}^n\min \{|x_i-y_i|,q-|x_i-y_i|\}.$$

$$S(n,e,q)=\{\mathbf{x}\in (\mathbb{Z}/q\mathbb{Z}{)}^n:d(\mathbf{x},0)\le e\}$$

$$S(n,e)=\{\mathbf{x}\in {\mathbb{Z}}^n:|x_1|+|x_2|+\cdots +|x_n|\le e\}$$

$$\{0\},\{\pm x_i{\}}_{1\le i\le n},\{\pm 2x_i{\}}_{1\le i\le n},\{\pm x_i\pm x_j{\}}_{1\le i<j\le n}$$

$$\displaystyle\sum_{i=1}^{n}\bigl{(}$$

$$\displaystyle\qquad+\sum_{1\leq i<j\leq n}\bigl{(}(x_{i}+x_{j})^{2k}+(x_{i}-x_{j})^{2k}+(-x_{i}+x_{j})^{2k}+(-x_{i}-x_{j})^{2k}\bigr{)}$$

$$=2(4^k+1)\sum _{i=1}^n{x}_i^{2k}+\sum _{1\le i<j\le n}4\sum _{t=0}^k\left(\genfrac{}{}{0ex}{}{2k}{2t}\right)x_i^{2t}{x}_j^{2(k-t)}$$

$$=(2^{2k+1}+4(n-1)+2)\sum _{i=1}^n{x}_i^{2k}+4\sum _{t=1}^{k-1}\sum _{1\le i<j\le n}\left(\genfrac{}{}{0ex}{}{2k}{2t}\right)x_i^{2t}{x}_j^{2(k-t)}$$

$$=(2^{2k+1}+4n-2)S_{2k}+2\sum _{t=1}^{k-1}\left(\genfrac{}{}{0ex}{}{2k}{2t}\right)(S_{2t}{S}_{2(k-t)}-S_{2k})$$

$$=(2^{2k}+4n+2)S_{2k}+2\sum _{t=1}^{k-1}\left(\genfrac{}{}{0ex}{}{2k}{2t}\right)S_{2t}{S}_{2(k-t)}$$

$$(4^k+4n+2)S_{2k}+2\sum _{t=1}^{k-1}\left(\genfrac{}{}{0ex}{}{2k}{2t}\right)S_{2t}{S}_{2(k-t)}=\{\begin{array}{ll}0& \text{if }p-1\nmid 2k,\\ -1& \text{if }p-1\mid 2k.\end{array}$$

$$X=\{ax+by:x\ge 1,y\ge 0\}.$$

$$0=(4^{k_0}+4n+2)S_{2k_0}+2\sum _{t=1}^{k_0-1}\left(\genfrac{}{}{0ex}{}{2k_0}{2t}\right)S_{2t}{S}_{2(k_0-t)}=(4^{k_0}+4n+2)S_{2k_0}$$

$$e_k=\sum _{1\le i_1<\cdots <i_k\le n}{x}_{i_1}^2{x}_{i_2}^2\cdots x_{i_k}^2$$

$$k_0{e}_{k_0}=e_{k_0-1}{S}_2-e_{k_0-2}{S}_4+\cdots +(-1{)}^{k_0-2}{e}_1{S}_{2(k_0-1)}+(-1{)}^{k_0-1}{S}_{2k_0}.$$

$$k_0{e}_{k_0}$$

$$=(-1{)}^{k_0-1}{S}_{2k_0}=0$$

Full text

1. Introduction

For an integer $q\geq 2$, consider the space $(\mathbb{Z}/q\mathbb{Z})^{n}$ equipped with the Lee metric $d$ given by

[TABLE]

An $e$-error-correcting Lee code is a subset $C\subseteq(\mathbb{Z}/q\mathbb{Z})^{n}$ such that any two distinct elements of $C$ have distance at least $2e+1$. An $e$-error-correcting Lee code $C$ is further called a perfect $e$-error-correcting Lee code if for each $x\in(\mathbb{Z}/q\mathbb{Z})^{n}$, there exists a unique element $c\in C$ such that $d(x,c)\leq e$. A perfect $e$-error-correcting Lee code in $(\mathbb{Z}/q\mathbb{Z})^{n}$ is also called simply a $PL(n,e,q)$-code.

There is an equivalent description of error-correcting Lee codes that uses the language of tilings. Consider the Lee sphere

[TABLE]

of radius $e$. An $e$-error-correcting Lee code is a subset $C\subseteq(\mathbb{Z}/q\mathbb{Z})^{n}$ such that for any $\mathbf{x}\neq\mathbf{y}$ in $C$, the two spheres $\mathbf{x}+S(n,e,q)$ and $\mathbf{y}+S(n,e,q)$ are disjoint. Thus it can be naturally identified with a translational packing of $S(n,e,q)$ in $(\mathbb{Z}/q\mathbb{Z})^{n}$. A perfect $e$-error-correcting Lee code then corresponds to a translational tiling of $(\mathbb{Z}/q\mathbb{Z})^{n}$ by $S(n,e,q)$.

If $q\geq 2e+1$, then the natural projection map $\mathbb{Z}^{n}\to(\mathbb{Z}/q\mathbb{Z})^{n}$ restricts to a bijection from

[TABLE]

to $S(n,e,q)$. Any tiling of $(\mathbb{Z}/q\mathbb{Z})^{n}$ by $S(n,e,q)$ will then pull back via the projection to a tiling of $\mathbb{Z}^{n}$ by $S(n,e)$. Let us call a subset $C\subseteq\mathbb{Z}^{n}$ a perfect $e$-error-correcting Lee code in $\mathbb{Z}^{n}$, or simply a $PL(n,e)$-code, if the translates of $S(n,e)$ centered at vectors of $C$ form a tiling of $\mathbb{Z}^{n}$. Then a $PL(n,e,q)$-code induces a $PL(n,e)$-code that is a disjoint union of cosets of $q\mathbb{Z}^{n}\subset\mathbb{Z}^{n}$. Conversely, any such $PL(n,e)$-code clearly comes from a $PL(n,e,q)$-code. We restate this in the following proposition.

Proposition 1.1**.**

For $q\geq 2e+1$, there exists a natural bijection between $PL(n,e,q)$-codes and $PL(n,e)$-codes that is a union of cosets of $q\mathbb{Z}^{n}\subset\mathbb{Z}^{n}$, given by taking the image or the inverse image with respect to the projection map $\mathbb{Z}^{n}\to(\mathbb{Z}/q\mathbb{Z})^{n}$.

Thus to know all about $PL(n,e,q)$-codes, it suffices to study $PL(n,e)$-codes.

Error-correcting codes in the Lee metric have been first investigated by Golomb and Welch [2]. In the paper, they explicitly construct $PL(1,e,2e+1)$-codes, $PL(2,e,2e^{2}+2e+1)$-codes, and $PL(n,1,2n+1)$-codes. On the other hand, they conjecture the nonexistence of perfect Lee codes for other $n$ and $e$.

Conjecture 1.2**.**

For $n\geq 3$ and $e\geq 2$, there exist no $PL(n,e)$-codes.

The case when $e$ is “large” compared to $n$ is studied extensively in the literature. Golomb and Welch [2] proved using a compactness argument that for each $n\geq 3$, there exists a sufficiently large $\rho_{n}$ such that there exist no $PL(n,e)$-codes for each $e\geq\rho_{n}$. An effective form of this theorem, that $PL(n,e,q)$-codes do not exist for $3\leq n\leq 5,e\geq n-1,q\geq 2e+1$ and $n\geq 6,e\geq\frac{\sqrt{2}}{2}n-\frac{3}{4}\sqrt{2}-\frac{1}{2},q\geq 2e+1$, was subsequently shown by Post [8]. Lepistö [7] improved the bound asymptotically and obtained the following theorem.

Theorem 1.3**.**

For any $n,e,q$ satisfying $n<(e+2)^{2}/2.1$ and $e\geq 285$ and $q\geq 2e+1$, there exist no $PL(n,e,q)$-codes.

Another direction of approach is to focus on small $n$. Gravier, Mollard, and Payan [3] showed the nonexistence of $PL(3,e)$-codes by analyzing possible local configurations. Later a computer-based proof of the nonexistence of $PL(4,e)$-codes was given by Špacapan [9], and Horak [5] further extended the theorem to prove nonexistence of $PL(n,e)$-codes for $3\leq n\leq 5$ and $e\geq 2$. In recent years, the case $e=2$ has been investigated for reasonably small $n$. For $n=5,6$, Horak [4] showed that $PL(5,2)$-codes and $PL(6,2)$-codes do not exist, and Horak and Grosěk [6] further showed using a computer that for $7\leq n\leq 12$ there are no linear $PL(n,2)$-codes, i.e., $PL(n,2)$-codes that is a lattice in $\mathbb{Z}^{n}$.

In this note, we continue along this line and provide a number theoretic condition under which $PL(n,2)$-codes do not exist. In particular, we prove the following theorem.

Theorem 1.4**.**

Suppose $p=2n^{2}+2n+1$ is prime. Let $a$ be the smallest positive integer for which $p\mid 4^{a}+4n+2$ and $b$ be the smallest positive integer for which $p\mid 4^{b}-1$. (For convenience let $a=\infty$ if there is no $a$ with $p\mid 4^{a}+4n+2$.) If the equation $a(x+1)+by=n$ has no nonnegative integer solutions, then $PL(n,2)$-codes do not exist. For instance, there are no $PL(n,2)$-codes for $n=5,7,9,12,14,17,\ldots$.

To illustrate the strength of this theorem, we provide numerical data concerning the number of $n$ to which the theorem can be applied. As in Table 1, if $2n^{2}+2n+1$ is indeed prime, in most cases the second condition about the equation having no nonnegative solutions is also satisfied. It is reasonable to expect that there are infinitely many $n$ such that $2n^{2}+2n+1$ is prime, although it is far from being proved. This is a special case of the Bunyakovsky conjecture, and moreover the heuristics of the Bateman–Horn conjecture [1] expects there to be asymptotically $Cx/\log x$ such $n\leq x$ for some absolute constant $C$.

The condition $2n^{2}+2n+1=\lvert S(n,2)\rvert$ being prime is included in order to use a result that allows us to translate the tiling problem to a purely algebraic problem. The following theorem is proved in [10].

Theorem 1.5**.**

Let $T\subset\mathbb{Z}^{n}$ be a finite subset of prime size $p$, and suppose that $T-T\subset\mathbb{Z}^{n}$ generates $\mathbb{Z}^{n}$ as an abelian group. Then there exists a tiling of $\mathbb{Z}^{n}$ by translates of $T$ if and only if there exists a homomorphism $\phi:\mathbb{Z}^{n}\to\mathbb{Z}/p\mathbb{Z}$ that restricts to a bijection from $T$ to $\mathbb{Z}/p\mathbb{Z}$.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

2. Proof of Theorem 1.4

In this section, we let $2n^{2}+2n+1=p$ be a prime. Since $\mathbb{Z}^{n}$ is a free abelian group generated by the unit vectors $e_{1},\dots,e_{n}$, a homomorphism $\phi:\mathbb{Z}^{n}\to\mathbb{Z}/p\mathbb{Z}$ is determined uniquely by the values $x_{i}=\phi(e_{i})$ for $1\leq i\leq n$. Then $\phi$ restricting to a bijection from $S(n,2)$ to $\mathbb{Z}/p\mathbb{Z}$ is equivalent to the sets

[TABLE]

forming a partition of $\mathbb{Z}/p\mathbb{Z}$.

Suppose that such $x_{1},\dots,x_{n}\in\mathbb{Z}/p\mathbb{Z}$ exist. The sum of $2k$-th powers of all the elements is

[TABLE]

where we denote $S_{t}=\sum_{i=1}^{n}x_{i}^{t}$. On the other hand, this is the sum of the $2k$-th powers of all elements of $\mathbb{Z}/p\mathbb{Z}$. Thus

[TABLE]

Let $a$ and $b$ be the least positive integers satisfying $p\mid 4^{a}+4n+2$ and $p\mid 4^{b}-1$. Consider the set

[TABLE]

Note that the set $X$ is closed under addition. We now claim the following.

Lemma 2.1**.**

If $1\leq k<(p-1)/2$ is not in $X$, then $S_{2k}=0$.

Proof.

We prove by induction on $k$. Suppose $S_{2k}=0$ for all $k\leq k_{0}-1$ that is not in $X$. We now show that $S_{2k_{0}}=0$ if $k_{0}\notin X$. Assume $k_{0}$ is not in $X$. Since any $k$ for which $p\mid 4^{k}+4n+2$ is of the form $a+by$ and thus in $X$, we see that $p\nmid 4^{k_{0}}+4n+2$.

Moreover, because $k_{0}\notin X$ and $X$ is closed under addition, for each $t$ either $t$ or $k_{0}-t$ is not in $X$. From Equation 1 and the induction hypothesis it follows that

[TABLE]

in $\mathbb{Z}/p\mathbb{Z}$. Because $4^{k}+4n+2\neq 0$, we immediately obtain $S_{2k_{0}}=0$. ∎

Let

[TABLE]

be the elementary symmetric polynomials with respect to $x_{1}^{2},x_{2}^{2},\dots,x_{n}^{2}$. Using a similar argument, we prove the following lemma.

Lemma 2.2**.**

If $1\leq k\leq n$ is not in $X$, then $e_{k}=0$.

Proof.

We again prove by induction on $k$. Suppose $e_{k}=0$ for all $k\leq k_{0}-1$ not in $X$, and also assume $k_{0}\notin X$. The Newton identities on $x_{1}^{2},\dots,x_{n}^{2}$ can be written as

[TABLE]

Because $X$ is closed under addition and $k_{0}\notin X$, for each $0<t<k_{0}$ either $t\notin X$ or $k_{0}-t\notin X$. From Lemma 2.1 and the inductive hypothesis, it follows that either $e_{t}=0$ or $S_{2(k_{0}-t)}=0$. Therefore

[TABLE]

and thus $e_{k_{0}}=0$ since $k_{0}\neq 0$. ∎

We now note that $e_{n}=x_{1}^{2}\cdots x_{n}^{2}$. Since none of $x_{1},\dots,x_{n}$ is [math], the square of their product $e_{n}$ is also not [math], and hence $n\in X$. Thus by Theorem 1.5, $PL(n,2)$-codes exist only if $n\in X$. This finishes the proof of Theorem 1.4.

3. Acknowledgments

The author would like to express gratitude to Peter Horak, who introduced the author to the problem and provided helpful comments.