7 problems

Let $t\geq 1$ and $q\geq 2t(t+1)+1$. For a function $f$, consider the encoding … where $p\in[2t:\lfloor q/2\rfloor]$. The code is required to function-correct for the Lee weight fu…

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 i…

A Lee sphere in $\mathbb{R}^n$ is a cross-polytope-type region defined by a Lee radius, and a tiling means a covering of $\mathbb{R}^n$ by translates of such spheres with disjoint…

For integers $n,e$, let a $PL(n,e)$-code mean a perfect $e$-error-correcting code in the Lee metric on $\mathbb{Z}^n$. Golomb--Welch strong conjecture. There is no $PL(n,e)$-code f…

For integers $n,e,q$, let a $PL(n,e,q)$-code mean a perfect $e$-error-correcting code in the Lee metric on $\mathbb{Z}_q^n$. Golomb--Welch weak conjecture. There is no $PL(n,e,q)$-…

Let ${\mathbb C}(m,j)$ be the code defined earlier in the source, and let ${\mathbb C}_{2^j}^{{\cal H}_m}$ be the code associated with the Hadamard matrix ${\cal H}_m$ and modulus…

Let ${\cal W}$ be a weighing matrix of order $n$ and weight $w$, and let ${\mathbb C}^{\cal W}_s$ be the associated code; in the comparison under discussion, $s$ is the relevant mo…