On linear diameter perfect Lee codes with distance 6

In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with radius $r\ge 2$ and dimension $n\ge 3$. 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 $n\ge 3$ 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 $n\ge 3$, there is a linear $DPL(n,6)$ code if and only if $n=3,11$.