Weight distributions of cyclic codes of length tlmtlm

Let FqFq be a finite field with qq elements, ll an odd prime, and t,vt,v positive integers such that lv|(q−1)lv|(q−1) and gcd(t,l)=1gcd(t,l)=1. In this paper, we give a combinatorial result and use it to determine the weight distribution of a cyclic code of length tlmtlm with t|(q−1)t|(q−1), which is an open question in Yang et al. (2013). Moreover we compute the weight distributions of cyclic codes of length tlmtlm, where q≡3(mod4) and t=4t=4 or 88.