On multifold packings of radius-1 balls in Hamming graphs

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

17pp. V.3: revised, added: classification of small binary 2-fold 1-packings (Sect. Va), discussion of multifold perfect codes (Sect. Vb), lower bound on the size of multiple coverings (Sect. IVb) V.2: Sections about MDS codes and unitrades added, the proof in Section II rewritten and completed; other revisions

References in corpus (2)

• Perfect 2-colorings of Hamming graphs
• A generalized concatenation construction for q-ary 1-perfect codes

Cited by in corpus (5)

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• On degree-$3$ and $(n-4)$-correlation-immune perfect colorings of $n$-cubes
• The classification of orthogonal arrays OA(2048,14,2,7) and some completely regular codes