On Binary Cyclic Codes With Few Weights

Henk D. L. Hollmann () and Qing Xiang ()
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Henk D. L. Hollmann: Philips Research Laboratories
Qing Xiang: University of Delaware, Department of Mathematical Sciences

A chapter in Finite Fields and Applications, 2001, pp 251-275 from Springer

Abstract: Abstract Let $$ {C_{{{t_{0}}}}}{,_{{{t_{1}}...,{t_{r}}}}} $$ denote the binary cyclic code of length n = 2 m − 1 with defining zeros $$ {\alpha ^{{{t_{0}}}}},{\alpha ^{{{t_{1}}}}}...,{a^{{{t_{r}}}}} $$ , where α is a primitive element of GF(2 m ). Using the method in [8], we determine the weight distribution of the following cyclic codes. (i) $$ {C_{{1,{t_{1}},{t_{2}}}}}, $$ , where m = 2r + 1, t 1 = 2r + 1, t 2 = 2 r−1 + 1. (This code appeared in Research Problem 9.7 of MacWilhams and Sloane [14].) (ii) $$ {C_{{1,t,{t^{2}}}}}, $$ where m = 2r + l, t = l + 22r+1 (This code appeared in a conjecture of Chang, Gaal, Golomb, Gong, and Kumar [5].) (iii) Several cyclic codes in the paper of Van Lint and Wilson [12]. (iv) C 1,t, where $$ m = 2r,t = \sum\nolimits_{{i = 0}}^{r} {{2^{{ik}}}} $$ , gcd(m, k) = 1.

Keywords: Weight Distribution; Cyclic Code; Primitive Element; Weight Enumerator; Power Moment (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1007/978-3-642-56755-1_20

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