On the non-existence of linear perfect Lee codes: The Zhang-Ge condition and a new polynomial criterion

TL;DR

This paper proves the non-existence of linear perfect Lee codes for almost all error levels e using a generalized criterion, extending previous results and introducing a new polynomial-based non-existence test.

Contribution

It extends the non-existence results of linear perfect Lee codes to almost all e by using a generalized Lucas theorem and introduces a new polynomial criterion for lattice tiling non-existence.

Findings

Non-existence of linear e-perfect Lee codes for infinitely many dimensions when e has a digit 1 in base-3 (not in units place).

A new polynomial criterion for non-existence of certain lattice tilings based on a prime p and a tile B.

Recovery of Zhang and Ge's criterion for Lee balls when p=3.

Abstract

The Golomb-Welch conjecture (1968) states that there are no $e$-perfect Lee codes in $\mathbb{Z}^n$ for $n\geq 3$ and $e\geq 2$. This conjecture remains open even for linear codes. A recent result of Zhang and Ge establishes the non-existence of linear $e$-perfect Lee codes in $\mathbb{Z}^n$ for infinitely many dimensions $n$, for $e=3$ and $4$. In this paper we extend this result in two ways. First, using the non-existence criterion of Zhang and Ge together with a generalized version of Lucas' theorem we extend the above result for almost all $e$ (i.e. a subset of positive integers with density $1$). Namely, if $e$ contains a digit $1$ in its base-$3$ representation which is not in the unit place (e.g. $e=3,4$) there are no linear $e$-perfect Lee codes in $\mathbb{Z}^n$ for infinitely many dimensions $n$. Next, based on a family of polynomials (the $Q$-polynomials), we present a new criterion for the non-existence of certain lattice tilings. This criterion depends on a prime $p$ and a tile $B$. For $p=3$ and $B$ being a Lee ball we recover the criterion of Zhang and Ge.