On $q$-ary shortened-$1$-perfect-like codes

TL;DR

This paper proves the optimality of certain $q$-ary shortened Hamming codes, extends bounds for multifold packings, and demonstrates the existence of codes with specific parameters that are not derivable from perfect codes.

Contribution

It generalizes a 1998 result on code optimality, establishes bounds for multifold packings, and shows new code existence beyond shortening perfect codes.

Findings

Shortened Hamming codes are optimal for certain parameters.

Punctured Hamming code is an optimal $q$-fold packing with minimum distance 2.

Existence of $4$-ary codes with parameters of shortened $1$-perfect codes that are not shortened perfect codes.

Abstract

We study codes with parameters of $q$-ary shortened Hamming codes, i.e., $(n=(q^m-q)/(q-1), q^{n-m}, 3)_q$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold packings of radius-$1$ balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal $q$-fold packing with minimum distance $2$. Secondly, for every admissible length starting from $n=20$, we show the existence of $4$-ary codes with parameters of shortened $1$-perfect codes that cannot be obtained by shortening a $1$-perfect code. Keywords: Hamming graph, multifold packings, multiple coverings, perfect codes.