On linear diameter perfect Lee codes with diameter 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 (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 $n\ge 3$ 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 $n\ge 3$, there is a linear $DPL(n,6)$ code if and only if $n=3,11$.