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In this work, we find a form for the homogeneous weight over the ring Rk,m, using the related theoretical results from the literature. We then use the first order Reed-Muller codes to find a distance-preserving map that takes codes over Rk,m to binary codes. By considering cyclic, constacyclic and quasicyclic codes over Rk;m of different lengths for different values of k and m, we construct a considerable number of optimal binary codes that are divisible with high levels of divisibility. The codes we have obtained are also quasicyclic with high indices and they are all self-orthogonal when km ≥ 4. The results, which have been obtained by computer search are tabulated. [ABSTRACT FROM AUTHOR]

In this work we introduce a novel approach to find reversible codes over different alphabets, using so-called coterm polynomials and a module-construction. We obtain many optimal reversible codes with these constructions. In an attempt to apply the constructions to the DNA, we identify k -bases of DNA with elements in the ring R 2 k = F 2 [ u ] / ( u 2 k − 1 ) , and by using a form of coterm polynomials, we are able to solve the reversibility and complement problems in DNA codes over this ring. With a freedom on the choice of k we are able to embed any DNA code in a suitable ring, giving an algebraic structure to the DNA codes. We are also able to find reversible and reversible-complement codes that are not necessarily linear cyclic codes. [ABSTRACT FROM AUTHOR]

In this work, extension theorems are used for self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over F 2 m + u F 2 m for m = 1 , 2 . The duality and distance preserving Gray maps from F 4 + u F 4 to ( F 2 + u F 2 ) 2 and F 2 4 are used to obtain self-dual codes whose binary Gray images are [ 64 , 32 , 12 ] -extremal self-dual. An F 2 + u F 2 -extension is used and as binary images, 178 extremal binary self-dual codes of length 68 with new weight enumerators are obtained. Especially the first examples of codes with γ = 3 and many codes with the rare γ = 4 , 6 parameters are obtained. In addition to these, two hundred fifty doubly even self dual [ 96 , 48 , 16 ] -codes with new weight enumerators are obtained from four-circulant codes over F 4 + u F 4 . New extremal doubly even binary codes of lengths 80 and 88 are also found by the F 2 + u F 2 -lifts of binary four circulant codes and thus a lower bound on the number of non-isomorphic 3-(80, 16, 665) designs is modified. [ABSTRACT FROM AUTHOR]

In this work, quadratic residue codes over the ring F 2 + u F 2 + u 2 F 2 with u 3 = u are considered. A duality and distance preserving Gray map from F 2 + u F 2 + u 2 F 2 to F 2 3 is defined. By using quadratic double circulant, quadratic bordered double circulant constructions and their extensions self-dual codes of different lengths are obtained. As Gray images of these codes and their extensions, a substantial number of new extremal self-dual binary codes are found. More precisely, thirty two new extremal binary self-dual codes of length 68, 363 Type I codes of parameters [ 72 , 36 , 12 ] , a Type II [ 72 , 36 , 12 ] code and a Type II [ 96 , 48 , 16 ] code with new weight enumerators are obtained through these constructions. The results are tabulated. [ABSTRACT FROM AUTHOR]

In this paper skew cyclic codes over the the family of rings Fq+vFq with v2 = v are studied for the first time in its generality. Structural properties of skew cyclic codes over Fq + vFq are investigated through a decomposition theorem. It is shown that skew cyclic codes over this ring are principally generated. The idempotent generators of skew-cyclic codes over Fq and Fq+vFq have been considered for the first time in literature. Moreover, a BCH type bound is presented for the parameters of these codes. [ABSTRACT FROM AUTHOR]

We consider quasi-cyclic codes over the ring $$\mathbb{F }_2+u\mathbb{F }_2+v\mathbb{F }_2+uv\mathbb{F }_2$$ , a finite non-chain ring that has been recently studied in coding theory. The Gray images of these codes are shown to be binary quasi-cyclic codes. Using this method we have obtained seventeen new binary quasi-cyclic codes that are new additions to the database of binary quasi-cyclic codes. Moreover, we also obtain a number of binary quasi-cyclic codes with the same parameters as best known binary linear codes that otherwise have more complicated constructions. [ABSTRACT FROM AUTHOR]

A lift of binary self-dual codes to the ring R2 is described. By using this lift, a family of self-dual codes over R2of length 17 are constructed. Taking the binary images of these codes, extremal binary self-dual codes of length 68 are obtained. For the first time in the literature, extremal binary codes of length 68 with γ= 4 andγ= 6 in W68;2 have been obtained. In addition to these, six new codes withγ= 0 and fourteen new codes with γ= 2 in W68;2 have also been found. [ABSTRACT FROM AUTHOR]

Abstract: The structure of DNA is used as a model for constructing good error correcting codes and conversely error correcting codes that enjoy similar properties with DNA structure are also used to understand DNA itself. Recently, naturally four element sets are used to model DNA by some families of error correcting codes. Hence the structure of such codes has been studied. In this paper, the authors first relate DNA pairs with a special 16 element ring. Then, the so-called cyclic DNA codes of odd length that enjoy some of the properties of DNA are studied. Their algebraic structure is determined. Further, by introducing a map, a family of cyclic codes over this ring is mapped to DNA codes. Hamming minimum distances are also studied. The paper concludes with some DNA examples obtained via this family of cyclic codes. [Copyright &y& Elsevier]

Cyclic codes over an infinite family of rings are defined. The general properties of cyclic codes over these rings are studied, in particular nontrivial one-generator cyclic codes are characterized. It is also proved that the binary images of cyclic codes over these rings under the natural Gray map are binary quasi-cyclic codes of index 2. Further, several optimal or near optimal binary codes are obtained from cyclic codes over R via this map. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we study construction techniques of formally self-dual codes over the infinite family of rings . These codes give rise to binary formally self-dual codes. Using these constructions, we obtain a number of good formally self-dual binary codes including even formally self-dual binary codes of parameters , , and odd formally self-dual binary codes of parameters , all of which have better minimum distances than the best known self-dual codes of the same lengths. [Copyright &y& Elsevier]

In this work, we establish a strong connection between group rings and self-dual codes. We prove that a group ring element corresponds to a self-dual code if and only if it is a unitary unit. We also show that the double-circulant and four-circulant constructions come from cyclic and dihedral groups, respectively. Using groups of order 8 and 16 we find many new construction methods, in addition to the well-known methods, for self-dual codes. We establish the relevance of these new constructions by finding many extremal binary self-dual codes using them, which we list in several tables. In particular, we construct 10 new extremal binary self-dual codes of length 68. [ABSTRACT FROM AUTHOR]

Centraliser codes are codes of length n 2 defined as centralisers of a given matrix A of order n . Their dimension, parity-check matrices, syndromes, and automorphism groups are investigated. A lower bound on the dimension is n , the order of A . This bound is met when the minimal polynomial is equal to the annihilator, i.e. for so-called cyclic (a.k.a. non-derogatory) matrices. If, furthermore, the matrix is separable and the adjacency matrix of a graph, the automorphism group of that graph is shown to be abelian and to be even trivial if the alphabet field is of even characteristic. [ABSTRACT FROM AUTHOR]