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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]

In this paper, we study quadratic residue (QR) codes of prime length q over the ring R = p + u p + v p with u 2 = u , v 2 = v and u v = v u = 0 , where p and q are distinct odd prime numbers. We analyze some basic properties of cyclic codes of length n over R , we define QR codes by their generating idempotents. Further, we discuss the extended QR codes. We present a considerable number of good p -ary codes as Gray images of QR codes over p + u p + v p by considering the case when p is an odd prime. [ABSTRACT FROM AUTHOR]

In this paper, we study hulls of cyclic codes of length 2 n (i.e., oddly even length) over the ring Z 4 of integers modulo 4 by viewing these codes as ideals of the quotient ring Z 4 [ x ] / < x 2 n − 1 > , where n is an odd positive integer. We express the generator polynomials of the hull of each cyclic code C of length 2 n over Z 4 in terms of the generator polynomials of the code C , and we note that the 2-dimension of the hull of the code C is an integer Θ satisfying 0 ≤ Θ ≤ 2 n. We also obtain an enumeration formula for cyclic codes of length 2 n over Z 4 with hulls of a given 2-dimension. Besides this, we study the average 2-dimension, denoted by E (2 n) , of the hulls of cyclic codes of length 2 n over Z 4 and establish a formula for E (2 n) that handles well. With the help of this formula, we deduce that E (2 n) = 5 n 6 when n ∈ N 2 , and we study the growth rate of E (2 n) with respect to the length 2 n when n ∉ N 2 , where N 2 denotes the set of all positive integers ω such that w divides 2 i + 1 for some positive integer i. Further, when n ∉ N 2 , we derive lower and upper bounds on E (2 n) and show that these bounds are attained at n = 7. We also illustrate these results with some examples. As an application of these results, we construct some entanglement-assisted quantum error-correcting codes (EAQECCs) over Z 4 with specific parameters from cyclic codes of oddly even length over Z 4. [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]