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In this study, we determine the structure of cyclic codes over the ring Z q [ u ] / 〈 u 2 〉 which is isomorphic to R = Z q + u Z q where q = p s , p is a prime, s is a positive integer, and u 2 = 0. This is equivalent to determining the algebraic structure of ideals of the polynomial quotient ring R [ x ] / 〈 x n − 1 〉 , which is addressed in this paper completely. By establishing the structure of ideals of R [ x ] / 〈 x n − 1 〉 with gcd ⁡ (p , n) = 1 , we present an exact formula that enumerates the number of ideals of this ring that leads to the enumeration of cyclic codes over this ring. Finally, we consider and explore some special families of cyclic codes for some specific q and determine their size. [ABSTRACT FROM AUTHOR]

Recently some special type of mixed alphabet codes that generalize the standard codes has attracted much attention. Besides Z 2 Z 4 -additive codes, Z 2 Z 2 [ u ] -linear codes are introduced as a new member of such families. In this paper, we are interested in a new family of such mixed alphabet codes, i.e., codes over Z 2 Z 2 [ u 3 ] where Z 2 [ u 3 ] = { 0 , 1 , u , 1 + u , u 2 , 1 + u 2 , u + u 2 , 1 + u + u 2 } is an 8-element ring with u 3 = 0 . We study and determine the algebraic structures of linear and cyclic codes defined over this family. First, we introduce Z 2 Z 2 [ u 3 ] -linear codes and give standard forms of generator and parity-check matrices and later we present generators of both cyclic codes and their duals over Z 2 Z 2 [ u 3 ] . Further, we present some examples of optimal binary codes which are obtained through Gray images of Z 2 Z 2 [ u 3 ] -cyclic codes. [ABSTRACT FROM AUTHOR]

In this paper, we study the algebraic structure of-additive codes which are-submodules whereis prime, andandare positive integers.-additive codes naturally generalizeand-additive codes which have been introduced recently. The results obtained in this work generalize a great amount of the studies done on additive codes. Especially, we determine the standard forms of generator and parity-check matrices for this family of codes. This leads to the identification of the type and the cardinalities of these codes. Furthermore, we present some bounds on the minimum distance and examples that attain these bounds. Finally, we present some illustrative examples for some special cases. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the lifted polynomials which generate reversible codes overFq, a finite field withqelement. Lifted polynomials are introduced by the authors Oztas and Siap [Lifted polynomials over F16and their applications to DNA codes, Filomat 27(3) (2013), pp. 459–466] overF16. Lifted polynomials have proven to be very advantageous. They are easy to construct and they can be used to construct codes with specific properties such as dimension and the length of codes. We also generalize the 4k-lifted polynomials which lead to reversible and reversible complement DNA codes over. Further we construct examples of codes overF8,F9,F16andF256that have the best possible parameters or attain the Griesmer bound, hence they are optimal codes generated by lifted polynomials. [ABSTRACT FROM PUBLISHER]

In this paper, we study \BBZ2\BBZ4-additive cyclic codes. These codes are identified as \BBZ4[x]-submodules of the ring Rr,s=\BBZ2[x]/\langle x^{r}-1\rangle\times\BBZ4\left[x\right]/\langle x^{s}-1\rangle. The algebraic structure of this family of codes is studied and a set of generator polynomials for this family as a \BBZ4[x]-submodule of the ring Rr,s is determined. We show that the duals of \BBZ2\BBZ4-additive cyclic codes are also cyclic. We also present an infinite family of Maximum Distance separable with respect to the singleton bound codes. Finally, we obtain a number of binary linear codes with optimal parameters from the \BBZ2\BBZ4-additive cyclic codes. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of linear codes over the non chain ring $$Z_{3}[v]/\langle v^{3}-v\rangle $$ . In order to study the codes, we first study the structure of this ring via a distance preserving Gray map which also induces a relation between codes over this ring and ternary codes. Further, the algebraic structure of cyclic and dual codes is also studied. A MacWilliams type Identity between the Gray weight enumerators of the original code and its dual is established. In all cases examples that illustrate the theorems and lemmas are provided. Also, a BCH-type bound and an example that attains this bound is presented. [ABSTRACT FROM AUTHOR]

In this paper, we obtain bounds for error correcting codes of some particular homogeneous weights over the ring of integers modulo $${2^l( {l\ge 3})}$$. We also study these bounds to check for perfect linear codes and have shown the non-existence of one error correcting perfect codes with respect to the homogeneous weight. [ABSTRACT FROM AUTHOR]

Following the very recent studies on \mathbb Z2\mathbb Z4 -additive codes, \mathbb Z2\mathbb Z2[u] -linear codes have been introduced by Aydogdu et al. In this paper, we introduce and study the algebraic structure of cyclic, constacyclic codes and their duals over the R -module \mathbb {Z}_{2}^\alpha R^\beta where R=\mathbb {Z}_{2}+u\mathbb {Z}_{2}=\left \{{0,1,u,u+1}\right \} is the ring with four elements and u^2=0 . We determine the generating independent sets and the types and sizes of both such codes and their duals. Finally, we present a bound and an optimal family of codes attaining this bound and also give some illustrative examples of binary codes that have good parameters which are obtained from the cyclic codes in \mathbb Z_2^\alpha R^\beta . [ABSTRACT FROM AUTHOR]