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In this paper, we construct a class of Z 4 Z 4 Z 4 -additive cyclic codes generated by 3-tuples of polynomials. We discuss their algebraic structure and show that generator matrices can be constructed for all codes in this class. We study asymptotic properties of this class of codes by using a Bernoulli random variable. Moreover, let 0 < δ < 1 be a real number such that the entropy h 4 ((k + l + t) δ 6) < 1 4 , we show that the relative minimum distance converges to δ and the rate of the random codes converges to 1 k + l + t , where k, l, and t are pairwise co-prime positive odd integers. Finally, we conclude that the Z 4 Z 4 Z 4 -additive cyclic codes are asymptotically good. [ABSTRACT FROM AUTHOR]

For any odd prime $p$ , and a divisor $\ell $ of $p$ , we consider $I_\ell $ to be the set of all divisors of $p-1$ , which are less than or equal to $\ell $. In this letter, we construct quantum codes from cyclic codes over $\mathbb F_{p}$ and $\mathbb F_{p} S_\ell $ , where $S_\ell =\prod _{i\in I_\ell }R_{i}$ , for $R_{i}=\frac {\mathbb F_{p}[u]}{\langle u^{i+1}-u\rangle }$. For that, first we construct linear codes and a Gray map over $R_\ell $. Using this construction, we study cyclic codes over $R_\ell $ , and then extend that over $\mathbb F_{p} S_\ell $. We also give a Gray map over $\mathbb F_{p} S_\ell $. Then, using necessary and sufficient condition of dual containing property for cyclic codes, we construct quantum MDS codes. It is observed that the quantum codes constructed are new in the literature. [ABSTRACT FROM AUTHOR]

Let R = F 4 + u F 4 , with u 2 = u and S = F 4 + u F 4 + v F 4 , w i t h u 2 = u , v 2 = v , u v = v u = 0 . In this paper, we study F 4 R S -cyclic codes of block length (α , β , γ) and construct cyclic DNA codes from them. F 4 R S -cyclic codes can be viewed as S [ x ] -submodules of F q [ x ] 〈 x α − 1 〉 × R [ x ] 〈 x β − 1 〉 × S [ x ] 〈 x γ − 1 〉 . We discuss their generator polynomials as well as the structure of separable codes. Using the structure of separable codes, we study cyclic DNA codes. By using Gray maps ψ 1 from R to F 4 2 and ψ 2 from S to F 4 3 , we give a one-to-one correspondence between DNA codons of the alphabets { A , T , G , C } 2 , { A , T , G , C } 3 and the elements of R , S , respectively. Then we discuss necessary and sufficient conditions of cyclic codes over F 4 , R, S and F 4 R S to be reversible and reverse-complement. As applications, we provide examples of new cyclic DNA codes constructed by our results. [ABSTRACT FROM AUTHOR]