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Báo cáo toán học: "On a Class of Constant Weight Codes"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: On a Class of Constant Weight Codes. | On a Class of Constant Weight Codes Mihai Caragiu Institute of Mathematics Bucharest and Department of Mathematics Pennsylvania State University E-mail caragiu@math.psu.edu Submitted July 31 1995 Accepted January 2 1996 Abstract. For any odd prime power q we first construct a certain non-linear binary code C q 2 having q2 q 2 codewords of length q and weight q 1 2 each for which the Hamming distance between any two distinct codewords is in the range q 2 3pq 2 q 2 3pq 2 that is almost constant . Moreover we prove that C q 2 is distance-invariant. Several variations and improvements on this theme are then pursued. Thus we produce other classes of binary codes C q n n 3 of length q that have almost constant weights and distances and which for fixed n and big q have asymptotically qn n codewords. Then we prove the possibility of extending our codes by adding the complements of their codewords. Also by using results on Artin L series it is shown that the distribution of the 0 s and 1 s in the codewords we constructed is quasi-random. Our construction uses character sums associated with the quadratic character  of Fq in which the range of summation is Fq. Relations with the duals of the double error correcting BCH codes and the duals of the Melas codes are also discussed. 1991 AMS Subject Classification Primary 11T71 Secondary 11T23 94B27 Typeset by Ạ S-TpX THE ELECTRONIC JOURNAL OF COMBINATORICS 3 1996 R.4 2 1. Introduction In the present paper we shall hrst construct for any odd prime power q a nonlinear constant weight code C q 2 with q2 q 2 codewords with the property that each nonzero distance lies in the interval q 2 q 3 2 2 In constructing such codes we shall use character sums associated with the quadratic character  of Fq2 in which the range of summation is Fq. Sums of this type were considered for example by Davenport 5 . He shows for example that if 0 is any element generating the hnite held Fpk over its prime subheld Fp and if  is the quadratic character .