lSlT 1997, Ulm. Germany, June 29 - July 4 On the Minimal Words of the Primitive BCH Codes Yuri Borissov, Nikolai L. Manev Institute of Mathematics, Bulgarian Academy of Sciences, 8 G.Bonchev str., Sofia 1113, Bulgaria Abstract - The number of minimal codewords of weight 10 and 11 in the primitive double-error- correcting BCH [am - 1, 2m - 2m - 1, 51 code and one of weight 12 in the extended code are determined. I. INTRODUCTION For the first time the sets of minimal codewords of lin- ear codes were studied in connection with a decoding algo- rithm by Tai-Yang Hwang [2]. Additional interest to them was generated by J. Massey [3] who used them to specify the access structure of a secret-sharing scheme. For definitions of a secret-sharing scheme and access structure determined by linear code we refer to [3, 11. Definition.[3] Let C be a binary code. A codeword c E C is called minimal if its support does not contain the support of other codeword as true subset. Here are some of the basic properties of the minimal code- words: Proposition [Z] Let C be a linear [n, I;. d] code. Then - If c is a codeword of weight wt(c) satisfying inequality - If c is minimal, then wt(c) 5 n - I; + 1. d 5 wt(c) 5 2d - 1, then c is minimal. The cardinalities of the complete set of minimal codewords for q-ary Hamming code and for RM(2,m) are obtained in Herein we consider the primitive BCH [am - 1, 2" - 2m - 1, 51 code C correcting two errors, its extended and their duals codes. The first weight for which there exist both mini- mal and_non-minimal codewords is 10 for C and, respectively, 12 for C. We determine the cardinalities of the sets of minimal (non- minimal) codewords of weights 10, 11 in C and 12 in e. They are all possible weights of minimal codewords in case m = 5. Theorem 1. Let C be the double-error-correcting BCH [Zm - 1, 2m - 2m - 1, 51 code. If m = 21+ 1 - odd, then the number of minimal codewords of weight 10 is PI. n(n - l)(n - 7)(n - 17)(nZ - 16n + 135) 2.1202 PIO = Aro - If m = 22 - even, then the number P10 of minimal codewords of weight 10 is equal to A0 = A10 - where ~IZ is the number of minimal codewords of weight 12 in the extended BCH code e. The theorem is a consequence of thefollowing lemma: Lemma If the extended BCH code C : [am, k, d + 11 has Szj non-minimal codewords of weight 2j, then the BCH code C : [2m - 1, k, d] has 92j-1 and sz3 non-minimal codewords of weights 2j - 1 and 2j, respectively, where A The expressions for P~z are simillar to ones in Theorem 1 (but longer and more complex to be placed in this column). The methods of proving results are based on counting code- words with fixed nonzero positions and ideas first used by Gorenstein,Peterson and Zierler [4]. In the case m odd the theorems can be proved also in a way involving design theory. ACKNOWLEDGEMENTS This research was partially supported by the Bulgarian NSF under Contract I-506/95. - 5)(n4 - 323n3 + 394n2 - 2008n + 4861) 144 200 where A10 is the number of codewords of weight 10. Theorem 2. The number of minimal codewords of weight I1 in the double-error correcting primitive BCH [2" - 1, 2" - 2m - 1, 51 code C is REFERENCES AshilcRmin A. and Barg S., "Combinatorial Aspects of Se- cret Sharing with Codes", Fourth International Workshop on ACCT'94, Novgorod, Russia (1994) pp 8-11. Td-Yang Ywang, Decoding linear block codes for minimizing word error rate, IEEE Trans. on Information Theory, IT-25, no.6, 733-737. Massey J., Minimal Codewords and Secret Sharing", in Proc. Sixth Joint Swedish-Russian Workshop on Inf. Theory, Molle, Sweden (1993) pp. 246-249. Gorenstein D.,Peterson W. and Zierler N.,Two-error correcting BCH codes are quasi-perfect, Inform. Contr.,vol.3, pp. 291-294, 1960. Authorized licensed use limited to: CBNU. Downloaded on June 18,2010 at 04:32:49 UTC from IEEE Xplore. Restrictions apply. View publication stats View publication stats