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Cohen, “The threshold probability of a code,” IEEE Trans. Inform. Theory, vol. 41, no. 2, pp. 469477, Mar. 1995. Reading, MA: Addison-Wesley, 1991. TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 1, JANUARY 1997 Optimal Linear Codes of Dimension 4 over F5 Iliya Boukliev, Stoyan Kapralov, Tatsuya Maruta, and Masaharu Fukui Abstract-Let n,(lc, d) be the smallest integer n for which there exists a linear code of length n, dimension IC, and minimum distance d, over a field of q elements. In this correspondence we determine n5 (4, d) for all but 22 values of d. Index Terms-Optimal q-ary linear codes, minimum-lengthbounds. I. INTRODUCTION Let F,” be the n-dimensional vector space over the Galois field Fq. The Hamming distance between two vectors of F: is defined to he the number of coordinates in which they differ. A q-ary linear [n, k, d; &ode is a k-dimensional linear subspace of F: with minimum distance d. Let n,(k, d) denote the smallest value of n for which there exists an [n, k, d; q]-code . An [n,(k, d), k, d; q]-code is called optimal. The Griesmer bound [Ill, [30] provides an important lower bound on n,(k,d) For given q and k this bound is attained for all sufficiently large values of d [15], [7], [17]. The values of nq(k, d) are completely determined (i.e., for all d) only for small values of q and k. For binary codes, n2(k, d) is known for k 5 7 [31]. For ternary codes the results of [20], [9], [161, [21, [231, [%I complete the determination of n3{k,d) for k 5 5. The exact values of n.+(k,d) for k 5 4 are determined in [I], [lo], [19], [24], [26]. The values of ns(k,d) are known for k 5 3 for all d [17]. In this correspondence we study optimal linear codes of dimension 4 over Fs. We solve the problem of finding n5(4, d) for all but 22 values of d. In Sections I1 and I11 we give lower bounds on 125 (4, d) which improve the Griesmer bound, i.e., the nonexistence of some four-dimensional codes over F5. In Section IV we give upper bounds on n5(4, d), i.e., the existence results. Some necessary prelimmanes and known results are given in each of these sections as well. Combining all known results we produce the table on 125 (4, d) which is given in the Appendix. 11. LOWER BOUNDS ON %5 (4, d) Let C’ he an [~l,k,d;g]-code and B2 denote the number of codewords of weight I in its dual code Ci. Manuscript received May 24, 1995; revised January 31, 1996. The work of I. Bouliliev and S. Kapralov was Supported in part by the Bulgarian National Science Fund under Grant I-407/1994 and Grant MM-502/1995. The material in this correspondence was presented in parr at the 4th International Workshop on Algebraic and Combinatorial Coding Theory, ACCT-4. Novgorod, Russia, September 11-17, 1994. 1. Bouklliev is with the Institute of Mathematics, Bulgarian Academy of Sciences, P.O.Box 323, 5000 V. Tarnovo, Bulgaria. S. Kapralov is with the Department of Mathematics, Technical University, 5300 Gabrovo, Bulgaria. T. Maruta is with the Meijo University, Junior College Division, Tenpaku Nagoya, 468 Japan. M. Fukui is with the Department of Mathematics, Meijo University, Tenpaku Nagoya, 468 Japan. Publisher Item Identifier S 0018-9448(97)00108-9. 0018-9448/97$10.00 0 1997 IEEE Authorized licensed use limited to: Nedyalko Nedyalkov. Downloaded on June 7, 2009 at 10:48 from IEEE Xplore. Restrictions apply.