Multifold 1-perfect codes

12/07/2022

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by   Denis S. Krotov, et al.

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A μ-fold (multifold) 1-perfect code 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 graph, where q is a prime power, we characterise all parameters of multifold 1-perfect codes and all parameters of additive multifold 1-perfect codes. Additionally, a description of parameters of additive 4-ary completely regular codes of covering radius 4 is given. Keywords: perfect codes, multifold packing, multiple covering, list-decoding codes, spreads, multispreads.