On linear diameter perfect Lee codes with diameter 6

TL;DR

This paper investigates diameter perfect Lee codes, providing a counterexample to a longstanding conjecture and characterizing when linear codes exist for certain dimensions and diameters.

Contribution

It disproves the conjecture that no diameter perfect Lee codes exist beyond certain parameters and identifies specific dimensions where linear codes are possible.

Findings

Counterexample to Horak and AlBdaiwi's conjecture.

Linear DPL(n,6) codes exist only for n=3 and n=11.

Provides new insights into the structure of diameter perfect Lee codes.

Abstract

In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with radius $r\ge2$ and dimension $n\ge3$. A diameter perfect code is a natural generalization of the perfect code. In 2011, Etzion (IEEE Trans. Inform. Theory, 57(11): 7473--7481, 2011) proposed the following problem: Are there diameter perfect Lee (DPL, for short) codes with diameter greater than four besides the $DPL(3,6)$ code? Later, Horak and AlBdaiwi (IEEE Trans. Inform. Theory, 58(8): 5490--5499, 2012) conjectured that there are no $DPL(n,d)$ codes for dimension $n\ge3$ and diameter $d>4$ except for $(n,d)=(3,6)$. In this paper, we give a counterexample to this conjecture. Moreover, we prove that for $n\ge3$, there is a linear $DPL(n,6)$ code if and only if $n=3,11$.