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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 study the structure and properties of additive right and left polycyclic codes induced by a nonbinary vector a ∈ F 4 n , where F 4 is the finite field of order 4. We show that additive right and left polycyclic codes are F 2 [ x ] -submodules of the rings R n = F 4 x / x n - a x and S n = F 4 x / x n - a r x respectively. We also show that these codes are invariant under multiplication by a certain matrix D, and construct their generator polynomials. Moreover, we study the relationship between additive polycyclic codes and linear polycyclic codes. We identify cases in which additive right polycyclic codes are linear right polycyclic codes and other cases in which additive right polycyclic codes are not linear right polycyclic codes. Finally, we give some applications of these codes by constructing examples of codes with good parameters. [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]

In this paper, we study the algebraic structure of additive cyclic codes over the alphabet F 2 r × F 4 s = F 2 r F 4 s , where r and s are non-negative integers, F 2 = GF (2) and F 4 = GF (4) are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for F 2 F 4 -additive cyclic codes. We also introduce a linear map W that depends on the trace map T to relate these codes to binary linear codes over F 2. Further, we investigate the duals of F 2 F 4 -additive cyclic codes. We show that the dual of any F 2 F 4 -additive cyclic code is another F 2 F 4 -additive cyclic code. Using the mapping W, we provide examples of F 2 F 4 -additive cyclic codes whose binary images have optimal parameters. We also consider additive cyclic codes over F 4 and give some examples of optimal parameter quantum codes over F 4 . [ABSTRACT FROM AUTHOR]