On Binary Cyclic Codes With Few Weights

ABSTRACT Let C t 0 ,t 1 ,⋯,t r denote the binary cyclic code of length n=2 m -1 with defining zeros α t 0 ,α t 1 ,⋯,α t r , where α is a primitive element of GF(2 m ). Using the method in [H. D. L. Hollmann and Q. Xiang, A proof of the Welch and Niho conjectures on cross-correlations of binary m-sequences, Finite Fields Appl. 7, 253-286 (2001; Zbl 1027.94006)], we determine the weight distribution of the following cyclic codes. (i) C 1,t 1 ,t 2 , where m=2r+1, t 1 =2 r +1, t 2 =2 r-1 +1. (This code appeared in Research Problem 9.7 of F. J. MacWilliams and N. J. A. Sloane [The theory of error-correcting codes, North-Holland (1977; Zbl 0369.94008)]). (ii) C 1,t,t 2 , where m=2r+1, t=1+2 r+1 . (This code appeared in a conjecture of A. Chang, P. Gaal, S. W. Golomb, G. Gong, and P. V. Kumar [On a sequence conjectured to have ideal 2-level auto-correlation function, ISIT 1998, Cambridge, MA (1998)]). (iii) Several cyclic codes in the paper of J. H. van Lint and R. M. Wilson [IEEE Trans. Inf. Theory 32, 23-40 (1986; Zbl 0616.94012)]. (iv) C 1,t , where m=2r, t=∑ i=0 r 2 ik , gcd(m,k)=1.