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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]

In this paper, we introduce double Quadratic Residue Codes (QRC) of length n = p + q for prime numbers p and q in the ambient space F 2 p × F 2 q. We give the structure of separable and non-separable double QRC over this alphabet and we show that interesting double QR codes in this space exist only in the case when p = q. We give the main properties for these codes such as their idempotent generators and their duals. We relate these codes to codes over rings and show how they can be used to construct interesting lattices. As an applications of these codes, we provide examples of self-dual, formally self-dual and optimal double QRC. We also provide examples of best known quantum codes that are derived from double-QRC in this setting. [ABSTRACT FROM AUTHOR]