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In this paper, we introduce F q 2 -double cyclic codes of length n = r + s , where F q 2 is the Galois field of q 2 elements, q is a power of a prime integer p and r, s are positive integers. We determine the generator polynomials for any F q 2 -double cyclic code. For any F q 2 -double cyclic code C , we will define the Euclidean dual code C ⊥ based on the Euclidean inner product and the Hermitian dual code C ⊥ H based on the Hermitian inner product. We will construct a relationship between C ⊥ and C ⊥ H and then find the generator polynomials for the Hermitian dual code C ⊥ H. As an application of our work, we will present examples of optimal parameter linear codes over the finite field F 4 and also examples of optimal quantum codes that were derived from F 4 -double cyclic codes using the Hermitian inner product. [ABSTRACT FROM AUTHOR]

In this paper, we give the exact number of Z 2 Z 4 -additive cyclic codes of length n = r + s , for any positive integer r and any positive odd integer s. We will provide a formula for the the number of separable Z 2 Z 4 -additive cyclic codes of length n and then a formula for the number of non-separable Z 2 Z 4 -additive cyclic codes of length n. Then, we have generalized our approach to give the exact number of Z p Z p 2 -additive cyclic codes of length n = r + s , for any prime p, any positive integer r and any positive integer s where gcd p , s = 1. Moreover, we will provide examples of the number of these codes with different lengths n = r + s . [ABSTRACT FROM AUTHOR]