A note on the weight distribution of some cyclic codes

Abstract

Let Fq be the finite field with q elements and Cn be the cyclic group of order n, where n is a positive integer relatively prime to q. Let H,K be subgroups of Cn such that H is a proper subgroup of K. In this note, the weight distributions of the cyclic codes of length n over Fq with generating idempotents K and eH,K=H-K are explicitly determined, where K=1/|K|∑gεKg and H=1/|H|∑gεHg. 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 pm over Fq, where p is an odd prime and q is a primitive root modulo pm. Finally, two examples are presented to illustrate our results.