On the complete weight enumerators of some reducible cyclic codes

Let FrFr be a finite field with r=qmr=qm elements and θθ a primitive element of FrFr. Suppose that h1(x)h1(x) and h2(x)h2(x) are the minimal polynomials over FqFq of g1−1 and g2−1, respectively, where g1,g2∈Fr∗. Let CC be a reducible cyclic code over FqFq with check polynomial h1(x)h2(x)h1(x)h2(x). In this paper, we investigate the complete weight enumerators of the cyclic codes CC in the following two cases: (1) g1=θq−1h,g2=βg1, where h>1h>1 is a divisor of q−1q−1, e>1e>1 is a divisor of hh, and β=θr−1e; (2) g1=θ2,g2=θpk+1g1=θ2,g2=θpk+1, where q=pq=p is an odd prime and kk is a positive integer. Moreover, we explicitly present the complete weight enumerators of some cyclic codes.