Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On optimal linear codes over F8. | On optimal linear codes over Fg Rie Kanazawa and Tatsuya Maruta Department of Mathematics and Information Sciences Osaka Prefecture University Sakai Osaka 599-8531 Japan maruta@ Submitted Aug 20 2010 Accepted Jan 29 2011 Published Feb 14 2011 Mathematics Subject Classification 94B05 94B27 51E20 05B25 Abstract Let nq k d be the smallest integer n for which there exists an n k d q code for given q k d. It is known that n8 4 d Vi 0 d 8i- for all d 833. As a continuation of Jones et al. Electronic J. Combinatorics 13 2006 R43 we determine n8 4 d for 117 values of d with 113 d 832 and give upper and lower bounds on n8 4 d for other d using geometric methods and some extension theorems for linear codes. 1 Introduction We denote by Fn the vector space of n-tuples over Fq the field of q elements. A q-ary linear code C of length n and dimension k an n k q code is a k-dimensional subspace of F . The Hamming distance d x y between two vectors x y G Fn is the number of nonzero coordinate positions in x y. The minimum distance of a linear code C is defined by d C min d x y x y G C x y which is equal to the minimum weight of C defined by wt C min wt x x G C x 0 where 0 is the all-0-vector and wt x d x 0 is the weight of x. A q-ary linear code of length n dimension k and minimum distance d is referred to as an n k d q code. The weight distribution of C is the list of numbers Ai which is the number of codewords of C with weight i. The weight distribution . for brevity with A0 Ad . 1 a . is also expressed as 01da . A k X n matrix having as rows the vectors of a basis of C is called a generator matrix of C. This research was partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 20540129. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P34 1 A fundamental problem in coding theory is to find nq k d the minimum length n for which an n k d q code exists 5 . An n k d q code is called optimal
