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Báo cáo toán học: "An Extremal Doubly Even Self-Dual Code of Length 112"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: An Extremal Doubly Even Self-Dual Code of Length 112. | An Extremal Doubly Even Self-Dual Code of Length 112 Masaaki Harada Department of Mathematical Sciences Yamagata University Yamagata 990-8560 Japan mharada@sci.kj.yamagata-u.ac.jp Submitted Dec 29 2007 Accepted Aug 24 2008 Published Aug 31 2008 Mathematics Subject Classifications 94B05 Dedicated to Professor Tatsuro Ito on His 60th Birthday Abstract In this note an extremal doubly even self-dual code of length 112 is constructed for the first time. This length is the smallest length for which no extremal doubly even self-dual code of length n 0 mod 24 has been constructed. 1 Introduction As described in 10 self-dual codes are an important class of linear codes for both theoretical and practical reasons. It is a fundamental problem to classify self-dual codes of modest length and determine the largest minimum weight among self-dual codes of that length. By the Gleason-Pierce theorem there are nontrivial divisible self-dual codes over Fq for q 2 3 and 4 only where Fq denotes the finite field of order q and this is one of the reasons why much work has been done concerning self-dual codes over these fields. A binary self-dual code C of length n is a code over F2 satisfying C C where the dual code C of C is defined as C x 2 Fn I x y 0 for all y 2 Cg under the standard inner product x y. A self-dual code C is doubly even if all codewords of C have weight divisible by four and singly even if there is at least one codeword of weight 2 mod 4 . Note that a doubly even self-dual code of length n exists if and only if n is divisible by eight. It was shown in 8 that the minimum weight d of a doubly even self-dual code of length n is bounded by d 4 n 24 4. A doubly even self-dual code meeting this upper bound is called extremal. The existence of extremal doubly even self-dual codes is known for the following lengths n 8 16 24 32 40 48 56 64 80 88 104 136 THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N33 1 and their existence was already known some 25 years ago see 7 Fig. 19.2