# On multifold packings of radius-1 balls in Hamming graphs

## TL;DR

This paper investigates multifold packings of radius-1 balls in Hamming graphs, providing bounds, classifications, and properties of such packings, with implications for error-correcting and list-decodable codes.

## Contribution

It introduces asymptotic bounds, characterizes optimal packings, and classifies small cases, advancing understanding of multifold packings in Hamming spaces.

## Key findings

- Asymptotic bounds for maximum size of 2-fold 1-packings as alphabet size grows
- Identification of optimal packings related to MDS codes when q ≥ 2n
- Classification of all optimal binary 2-fold 1-packings up to length 9

## Abstract

A $\lambda$-fold $r$-packing (multiple radius-$r$ covering) in a Hamming metric space is a code $C$ such that the radius-$r$ balls centered in $C$ cover each vertex of the space by not more (not less, respectively) than $\lambda$ times. The well-known $r$-error-correcting codes correspond to the case $\lambda=1$, while in general multifold $r$-packing are related with list decodable codes. We (a) propose asymptotic bounds for the maximum size of a $q$-ary $2$-fold $1$-packing as $q$ grows; (b) prove that a $q$-ary distance-$2$ MDS code of length $n$ is an optimal $n$-fold $1$-packing if $q\ge 2n$; (c) derive an upper bound for the size of a binary $\lambda$-fold $1$-packing and a lower bound for the size of a binary multiple radius-$1$ covering (the last bound allows to update the small-parameters table); (d) classify all optimal binary $2$-fold $1$-packings up to length $9$, in particular, establish the maximum size $96$ of a binary $2$-fold $1$-packing of length $9$; (e) prove some properties of $1$-perfect unitrades, which are a special case of $2$-fold $1$-packings. Keywords: Hamming graph, multifold ball packings, two-fold ball packings, list decodable codes, multiple coverings, completely regular codes, linear programming bound