135 research outputs found

    Some new results on the self-dual [120,60,24] code

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    The existence of an extremal self-dual binary linear code of length 120 is a long-standing open problem. We continue the investigation of its automorphism group, proving that automorphisms of order 30 and 57 cannot occur. Supposing the involutions acting fixed point freely, we show that also automorphisms of order 8 cannot occur and the automorphism group is of order at most 120, with further restrictions. Finally, we present some necessary conditions for the existence of the code, based on shadow and design theory.Comment: 23 pages, 6 tables, to appear in Finite Fields and Their Application

    On extremal self-dual ternary codes of length 48

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    All extremal ternary codes of length 48 that have some automorphism of prime order p≥5p\geq 5 are equivalent to one of the two known codes, the Pless code or the extended quadratic residue code

    The Automorphism Group of an Extremal [72,36,16] Code does not contain elements of order 6

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    The existence of an extremal code of length 72 is a long-standing open problem. Let C be a putative extremal code of length 72 and suppose that C has an automorphism g of order 6. We show that C, as an F_2-module, is the direct sum of two modules, one easily determinable and the other one which has a very restrictive structure. We use this fact to do an exhaustive search and we do not find any code. This proves that the automorphism group of an extremal code of length 72 does not contain elements of order 6.Comment: 15 pages, 0 figures. A revised version of the paper is published on IEEE Transactions on Information Theor

    Self-Dual Codes

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    Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

    Extremal doubly-even [80,40,16] codes with an automorphism of order 19

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    AbstractIn this paper, it is shown that there are 11 extremal doubly-even self-dual [80,40,16] codes with an automorphism of order 19, up to equivalence, and the codes are new. The automorphism groups of the codes are also obtained
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