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An enumeration of binary self-dual codes of length 32. (English) Zbl 1004.94027

The authors provide several algorithms to enumerate binary self-dual codes recursively. The first algorithm computes the \((2k,k,2)\) self-dual codes from the \((2j,j)\) self-dual codes \((j\leq k-1)\). The next one takes the list of \((2j,j,d\geq 4)\) self-dual codes \((j\leq k-2)\) and produces a list of all inequivalent \((2k,k,4)\) self-dual codes. The third algorithm takes a subset of the codes produced by the second algorithm to produce a list of inequivalent \((2k,k,d\geq 6)\) self-dual codes. As the algorithms also deliver the size of the automorphism groups for each code, it was possible to check that the list contains the correct number of codes. The paper concludes with a list of the weight enumerators and size of the automorphism groups for self-dual codes of length 32, extending results of J. H. Conway and V. Pless [J. Comb. Theory, Ser. A 28, 26-53 (1980; Zbl 0439.94011)] and J. H. Conway, V. Pless and N. J. A. Sloane [ibid. 60, 183-195 (1992; Zbl 0751.94009)] for the binary doubly-even self-dual codes of length 32. It also proves a conjecture of J. H. Conway and N. J. A. Sloane [ibid. 36, 1319-1333 (1990; Zbl 0713.94016)], which states that the smallest possible length for a binary self-dual code whose full automorphism group has size 1 is 34.
Reviewer: K.Roegner (Berlin)

MSC:

94B05 Linear codes (general theory)
05B05 Combinatorial aspects of block designs
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