A class of binary cyclic codes and sequence families

Abstract

For two odd integers l, k with \(0<l<k\) and \(\gcd (l,k)=1\), let \(m=2k\) and \(d=\frac{2^{lk}+1}{2^l+1}+\frac{2(2^m-1)}{3}\). In this paper, we determine the value distribution of the exponential sum \(\sum _{x\in \mathbb {F}_{2^m}}(-1)^{\mathrm {Tr}_1^m(ax+bx^d)}\). As applications, the weight distribution of a class of binary cyclic codes is settled. Second, we determine the correlation distribution among sequences in a sequence family.