Complete weight enumerators of some cyclic codes

Abstract

Let F-r be a finite field with r = q(m) elements, alpha a primitive element of F-r, Tr-r/q the trace function from F-r onto F-q, r - 1 = nN for two integers n, N >= 1, and theta = alpha(N). In this paper, we use Gauss sums to investigate the complete weight enumerators of irreducible cyclic codes C = {c(a) = (Tr-r/q (a), Tr-r/q (a theta), ..., Tr-r/q(a theta(n-1)) : a is an element of F-r} and explicitly present the complete weight enumerators of some irreducible cyclic codes when gcd(n, q - 1) = q - 1 or q - 1/2. Moreover, we determine the complete weight enumerators of a class of cyclic codes with the check polynomials h(1)(x)h(2)(x) by using Gauss sums, where h(i) (x) are the minimal polynomials of alpha(-1)(i) over F-q and F-qmi* = <alpha(i)> for i = 1, 2. We shall obtain their explicit complete weight enumerators if gcd(m(1), m(2)) = 1 and q = 3 or 4