Weight distributions of cyclic codes with respect to pairwise coprime order elements

Let F r be an extension of a finite field F q with r = q m . Let each g i be of order n i in F r * and gcd ? ( n i , n j ) = 1 for 1 ≤ i ? j ≤ u . We define a cyclic code over F q by C ( q , m , n 1 , n 2 , ? , n u ) = { C ( a 1 , a 2 , ? , a u ) : a 1 , a 2 , ? , a u ? F r } , where C ( a 1 , a 2 , ? , a u ) = ( Tr r / q ( ? i = 1 u a i g i 0 ) , ? , Tr r / q ( ? i = 1 u a i g i n - 1 ) ) and n = n 1 n 2 ? n u . In this paper, we present a method to compute the weights of C ( q , m , n 1 , n 2 , ? , n u ) . Further, we determine the weight distributions of the cyclic codes C ( q , m , n 1 , n 2 ) and C ( q , m , n 1 , n 2 , 1 ) .

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