Cyclic codes with few weights and Niho exponents

AbstractThis paper studies the values of the sums Sk(a)=∑x∈F2m(-1)Tr(xk+ax),a∈F2m,where Tr is the trace function on F2m, m=2t and gcd(2m-1,k)=1. We mainly prove that when k≡2j(mod2t-1), for some j, then Sk(a) takes at least four values when a runs through F2m. This result, and other derived properties, can be viewed in the study of weights of some cyclic codes and of crosscorrelation function of m-sequences

Recent progress on weight distributions of cyclic codes over finite fields

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions