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Subsets of defect 3 in elementary Abelian 2-groups. (English) Zbl 1090.94026
Summary: It is well-known [see Elwyn R. Berlekamp, Algebraic coding theory. Moskau: Verlag “Mir” (1971; Zbl 0256.94006)] that linear codes over a two-element field are precisely subgroups of an elementary Abelian 2-group \(G\). It is naturally to consider subsets in \(G\) which are close to subgroups, as codes which are close to linear ones. In this connection in B. V. Novikov [Discrete mathematics and applications. Proceedings of the 6th international conference, Bansko, Bulgaria, August 31-September 2, 2001. Blagoevgrad: South-West University. Res. Math. Comput. Sci., 97–101 (2002; Zbl 1062.20058)] the notion of a defect of a subset of a group \(G\) has been introduced as a measure of a deviation from a subgroup (so that a subset has the defect 0 only if it is a subgroup). The subsets of defect 1 and 2 are described by B. V. Novikov. In this description the so called standard subsets play a leading role (see definition in section 2): all subsets of defect 1 are standard, and among subsets of defect 2 there is only one non-standard. In this article we show that all subsets of defect 3 containing not less than 12 elements are standard, and we describe all non-standard ones.
94B05 Linear codes, general