On linear diameter perfect Lee codes with distance 6

Abstract

In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with radius r≥2 and dimension n≥3. A diameter perfect code is a natural generalization of the perfect code. In 2011, Etzion (2011) [5] proposed the following problem: Are there diameter perfect Lee (DPL, for short) codes with distance greater than four besides the DPL(3,6) code? Later, Horak and AlBdaiwi (2012) [12] conjectured that there are no DPL(n,d) codes for dimension n≥3 and distance d>4 except for (n,d)=(3,6). In this paper, we give a counterexample to this conjecture. Moreover, we prove that for n≥3, there is a linear DPL(n,6) code if and only if n=3,11.