TAILIEUCHUNG - Báo cáo toán học: "New Optimal Constant Weight Codes"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: New Optimal Constant Weight Codes. | New Optimal Constant Weight Codes I. Gashkov D. Taub Department of Mathematics Karlstad University Sweden Submitted May 18 2007 Accepted Jun 18 2007 Published Jun 21 2007 Mathematics Subject Classifications 94B60 Abstract In 2006 Smith et al. published a new table of constant weight codes updating existing tables originally created by Brouwer et al. This paper improves upon these results by filling in 9 missing constant weight codes all of which are optimal by the second Johnson bound. This completes the tables for A n 16 9 and A n 18 10 up to n 63 and corrects some A n 14 8 . Introduction A binary constant weight code is any subset of Bn such that all elements codewords have the same weight. An important problem in coding theory is finding A n d w the maximum possible number of codewords in a constant weight code with length n minimum distance d and weight w. A large table of lower bounds on these numbers was published by Brouwer et al. 1 and later added to by Smith et al. 4 . However these tables are still far from complete and many of the existing values can be improved upon. When the lower bound for a code matches the upper bound the code is said to be optimal. There are a number of different methods for determining upper bounds for constant weight codes but for the codes presented in this paper the most useful is the one given by theorem 1. Theorem 1 The second Johnson bound There can be a code with parameters n d and w and size M only if n b a a 1 ba a 1 w M M - 1 d 2 holds where a and b are the unique integers such that wM an b and 0 b n. The research was support by The Royal Swedish Academy of Sciences THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N13 1 Proof See 2 p. 526. The second Johnson bound J2 n d w is the largest M such that the inequality in theorem 1 holds. It is possible that the inequality holds for all M in which case J2 n d w 1. All of the codes presented in this paper are optimal by the second Johnson

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