28 problems

Let $n$, $r$, and $q$ be positive integers, and let a $q$-ary perfect $r$-error-correcting Lee code of length $n$ be a perfect $r$-error-correcting Lee code over an alphabet of siz…

Let $F$ be a finite alphabet of cardinality $q\geq 2$, let $n$ be a positive integer, and let $V=F^n$. For words $\mathbf{x},\mathbf{y}\in V$, define the Hamming distance…

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…

Let $F$ be a finite alphabet of cardinality $q \geq 2$, let $n$ be a positive integer, and let $V=F^n$. For words $\mathbf{x},\mathbf{y}\in V$, define their Hamming distance…

Consider the Clark–Liang association scheme $CL(15,2)$ on $X=\mathbb{Z}_{15}$, with relation classes determined by $X_1=C_1\cup C_7$, $X_2=C_3$, and $X_3=C_5$, and adjacency matric…

Nonexistence conjecture. Consequently, all parameters of perfect $e-$error correcting codes were found if $e \ge 3$, and it was conjectured that no perfect $2-$error correcting cod…

Let $K(n,r)$ be the Kneser graph whose vertices are the $r$-subsets of $[n]$, with two vertices adjacent when they are disjoint. A subset $C$ of the vertices of a graph is a perfec…

An additive $1$-perfect code $C$ is a submodule of $\mathbb{Z}_4^{2m+n”}\times\mathbb{Z}_2^{2n'}$, and the quotient $Q=(\mathbb{Z}_4^{2m+n”}\times\mathbb{Z}_2^{2n'})/C$ is isomorph…

Let codes be considered in the Niederreiter–Rosenbloom–Tsfasman metric for $s\geq 2$ chains of length $r\geq 2$, and let $R$-perfect codes mean perfect codes with covering radius…

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…

Let $q$ be a prime power and consider nonlinear codes with parameters … These are the parameter family discussed for shortened $1$-perfect codes. Generalization conjecture. For eve…

Consider partitions of $H(5,5)$ into $25$ $(5,5^3,3)_5$ MDS codes, and call a partition lengthenable if it can be extended to a partition of $H(6,5)$ into $25$ perfect codes. Scarc…

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 $PL(n,e,q)$-…

Let $Q_d$ be the $d$-dimensional hypercube, and let a dominating set be a set of vertices such that every vertex is either in the set or adjacent to a vertex in it. Suppose that…

Let a total perfect code be a code in a graph in which every vertex has exactly one neighbor in the code, and consider lattice-like total perfect codes in the lattice graph on…

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…

Negligible-fraction conjecture. The fraction of perfect codes that include Preparata codes is negligibly small.

Nonexistence conjecture. There are no binary $1$-error-correcting perfect codes other than the Hamming code.

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

Let $p=2$, and consider perfect codes in $\mathbb{Z}_q^n$ under the $\ell_2$ metric, where $q$ and $n$ are positive integers. Such codes may be used as inner codes in constructions…