A note on the weight distribution of some cyclic codes

Let FqFq be the finite field with q   elements and CnCn be the cyclic group of order n, where n is a positive integer relatively prime to q  . Let H,KH,K be subgroups of CnCn such that H is a proper subgroup of K. In this note, the weight distributions of the cyclic codes of length n   over FqFq with generating idempotents Kˆ and eH,K=Hˆ−Kˆ are explicitly determined, where Kˆ=1/|K|∑g∈Kg and Hˆ=1/|H|∑g∈Hg. Our result naturally gives a new characterization of a theorem by Sharma and Bakshi [18] that determines the weight distribution of all irreducible cyclic codes of length pmpm over FqFq, where p is an odd prime and q   is a primitive root modulo pmpm. Finally, two examples are presented to illustrate our results.