The prime-order perfect Lee code linearity conjecture

Let $PL(n,e,q)$ denote a perfect $e$-error-correcting Lee code over $\mathbb{Z}_q^n$. Assume that $2n+1$ is prime. Prime-order perfect Lee code conjecture. Every $PL(n,1,q)$-code is linear and is a periodic extension of the unique, up to congruence, $PL(n,1,2n+1)$-code. The paper reports that this conjecture has been proved for $n=2,3$ and $n=5$, while it remains open in general.