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In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring M k (R) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring M k (R) are one sided ideals in the group matrix ring M k (R) G and the corresponding codes over the ring R are G k -codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes. [ABSTRACT FROM AUTHOR]

We describe skew G-codes, which are codes that are the ideals in a skew group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. These codes generalize many of the well-known classes of codes such as cyclic, quasicyclic, constacyclic codes, skew cyclic, skew quasicyclic and skew constacyclic codes. Additionally, using the skew G-matrices, we can generalize almost all the known constructions in the literature for self-dual codes. [ABSTRACT FROM AUTHOR]

In this paper, we give a matrix construction method for designing DNA codes that come from group matrix rings. We show that with our construction one can obtain reversible G k -codes of length kn, where k , n ∈ N , over the finite commutative Frobenius ring R. We employ our construction method to obtain many DNA codes over F 4 that satisfy the Hamming distance, the reverse, the reverse-complement and the fixed GC-content constraints. Moreover, we improve many lower bounds on the sizes of some known DNA codes and we also give new lower bounds on the sizes of DNA codes of lengths 48, 56, 60, 64 and 72 for some fixed values of the Hamming distance d. [ABSTRACT FROM AUTHOR]

Abstract: For a commutative noetherian ring R, we investigate relations between tilting and cotilting modules in and , where runs over the maximal spectrum of R. For each , we construct a correspondence between (equivalence classes of) n-cotilting R-modules C and (equivalence classes of) compatible families of n-cotilting -modules ( ). It is induced by the assignment , where is the colocalization of C at , and its inverse . We construct a similar correspondence for n-tilting modules using compatible families of localizations; however, there is no explicit formula for the inverse. [Copyright &y& Elsevier]

In this work, we present a number of generator matrices of the form [ I 2 n | τ 2 (v) ] , where I 2 n is the 2 n × 2 n identity matrix, v is an element in the group matrix ring M 2 (R) G and where R is a finite commutative Frobenius ring and G is a finite group of order 18. We employ these generator matrices and search for binary [ 72 , 36 , 12 ] self-dual codes directly over the finite field F 2. As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings. [ABSTRACT FROM AUTHOR]