Abstract
LetG be a finite abelian group,t a positive integer. Thet-shift sphere with centerx ∈G is the setS t (x)={±ix|i=1,...,t}. At-shift code is a subsetX ofG such that the setsS t (x) (x ∈X) have size 2t and are disjoint. Clearly, the sphere packing bound: 2t|X|+1≤|G| holds for anyt-shift codeX. Aperfect t-shift code is at-shift codeX with 2t|X|+1=|G|. A necessary and sufficient condition for the existence of a perfectt-shift code in a finite abelian group is known fort-1, 2. In this paper, we determine finite abelian groups in which there exists a perfectt-shift code fort=3, 4.
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This research was completed during the author's visit at the Institute for System Analysis, Moscow, as a Heizaemon Honda fellow of the Japan Association for Mathematical Sciences.
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Munemasa, A. On perfectt-shift codes in abelian groups. Des Codes Crypt 5, 253–259 (1995). https://doi.org/10.1007/BF01388387
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DOI: https://doi.org/10.1007/BF01388387