13 problems

Let $B(n-1)$ denote the number of $1$-perfect binary codes of length $n-1=2^m-1$, and let $\widetilde K_{\scriptscriptstyle LA}(n)$ be the lower-bound expression satisfying…

Nonexistence conjecture. Let $q \ge 6$ be a non-prime power. Then, there are no perfect error correcting codes over $Z_q$.

Let $n\geq 3$, and let a Cayley sum graph of the symmetric group $S_n$ be connected. A subgroup total perfect code is a subgroup of $S_n$ that is a total perfect code in this graph…

Let $n\geq 3$, and let a Cayley sum graph of the symmetric group $S_n$ be connected. A subgroup perfect code is a subgroup of $S_n$ that is a perfect code in this graph. Subgroup p…

Generalized Fibonacci cube perfect-code conjecture. If $C$ is a perfect code in $\Gamma_n(1^s)$, then

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…

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 $PL(n,e,q)$-…

Linear-components conjecture. Every $i$-component of $C$, for every $i$, is linear; equivalently, it is equivalent to

Weight-support system conjecture. All codewords of weight $3$ of a perfect code containing a Preparata code, and all codewords of weight $4$ of an extended perfect code containing…

Conjecture. There are no linear perfect codes with parameters $(n,r,2)$ except when $r=1$ or

Cyclicity conjecture for Abelian planar difference sets. The period of $\mathcal L$ in $A_n$ along $\mathbf f_{i,j}$ is equal to $n^2+n+1$ for at least one vector $\mathbf f_{i,j}$…

Prime power conjecture. There exists a linear $1$-perfect code in $(A_n,d)$, equivalently in $(\mathbb{Z}^n,d_a)$, if and only if $n$ is a prime power.