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In this paper, we employ a virus optimization (VO) algorithm, which is one of the metaheuristic optimization techniques, and a known construction method to compute many new binary $ [72,36,12] $ self-dual codes and optimal/near-optimal linear complementary dual (LCD) codes. In particular, we obtain 39 Type Ⅰ and 19 Type Ⅱ codes of length 72, with parameters in their weight enumerators that were not known in the literature before, and 85 new binary LCD codes that are either optimal or near-optimal. We also present the generator matrix of extended Golay code $ [24,12,8] $ by a cyclic group matrix ring element. [ABSTRACT FROM AUTHOR]

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]

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]

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well. [ABSTRACT FROM AUTHOR]

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]