Multifold 1-perfect codes
Denis S. Krotov
TL;DR
We address the problem of characterizing multifold $1$-perfect codes in $q$-ary Hamming graphs $H(n,q)$, with $q$ a prime power, and show that additive code structures via multispreads provide the key to realizations beyond prime $q$. The authors introduce multispreads to link completely regular codes with covering radius $1$ to $(\lambda,\mu)$-spreads, and prove that the kernel of a suitable check matrix $M(T_1,\dots,T_n)$ yields an $\mathbb{F}_p$-linear code precisely when the $(T_i)$ form a $(\lambda,\mu)$-spread. They establish necessary and sufficient conditions for the existence of $\mu$-fold $1$-perfect codes in $H(n,q)$ for $q=p^t$, including Lloyd-type congruence constraints, and show sufficiency by constructing additive codes and unions of cosets. Consequently, every admissible parameter set can be realized as a union of cosets of an additive multifold $1$-perfect code, thereby extending known prime-power results to non-prime $q$ and providing a constructive framework for multifold packing and list-decoding codes. The work forges a bridge between spreads, multispreads, and completely regular codes and opens directions for further study of additive CR codes and radius-2 generalizations.
Abstract
A multifold $1$-perfect code ($1$-perfect code for list decoding) in any graph is a set $C$ of vertices such that every vertex of the graph is at distance not more than $1$ from exactly $μ$ elements of $C$. In $q$-ary Hamming graphs, where $q$ is a prime power, we characterize all parameters of multifold $1$-perfect codes and all parameters of additive multifold $1$-perfect codes. In particular, we show that additive multifold $1$-perfect codes are related to special multiset generalizations of spreads, multispreads, and that multispreads of parameters corresponding to multifold $1$-perfect codes always exist. Keywords: perfect codes, multifold packing, multiple covering, list-decoding codes, additive codes, spreads, multispreads, completely regular codes, intriguing sets.