On the complete weight enumerators of some reducible cyclic codes

Let F r be a finite field with r = q m elements and ? a primitive element of F r . Suppose that h 1 ( x ) and h 2 ( x ) are the minimal polynomials over F q of g 1 - 1 and g 2 - 1 , respectively, where g 1 , g 2 ? F r ? . Let C be a reducible cyclic code over F q with check polynomial h 1 ( x ) h 2 ( x ) . In this paper, we investigate the complete weight enumerators of the cyclic codes C in the following two cases: (1) g 1 = ? q - 1 h , g 2 = β g 1 , where h 1 is a divisor of q - 1 , e 1 is a divisor of h , and β = ? r - 1 e ; (2) g 1 = ? 2 , g 2 = ? p k + 1 , where q = p is an odd prime and k is a positive integer. Moreover, we explicitly present the complete weight enumerators of some cyclic codes.

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