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In this work, we present a matrix construction for reversible codes derived from skew dihedral group rings. By employing this matrix construction, the ring $ \mathcal{F}_{j, k} $ and its associated Gray maps, we show how one can construct reversible codes of length $ n2^{j+k} $ over the finite field $ \mathbb{F}_4. $ As an application, we construct a number of DNA codes that satisfy the Hamming distance, the reverse, the reverse-complement, and the GC-content constraints with better parameters than some good DNA codes in the literature. [ABSTRACT FROM AUTHOR]

We determine the number of symmetric dualities on the $ s $-fold cross product of the cyclic group of order $ 2, $ which is the additive group of the finite field $ {\mathbb{F}}_{2^s}. $ We show that the ratio of symmetric dualities over all dualities goes to $ 0 $ as $ s $ goes to infinity.We also prove a surprising result that given any two binary codes $ C $ and $ D $ of the same length $ n $ with $ |C||D| = 2^n $, then viewing them as groups there is a symmetric duality $ M $ with $ C^M = D $, which also relates their weight enumerators as additive codes in a group via the MacWilliams relations. Using this theorem we show that any additive code in this setting can be viewed as an additive complementary dual code of length $ 1 $ with respect to some duality. [ABSTRACT FROM AUTHOR]

We define additive skew G-codes over finite fields and discuss several dualities attached to these codes. We examine the properties of self-dual skew G-codes and in particular we show that the dual, under any duality, of an additive skew G-code is also an additive skew G-code. Additionally, we propose a matrix construction for additive skew G-codes and use it to construct several examples of extremal self-dual additive skew G-codes over the finite field F 4 . Such codes have a strong connection to quantum error correcting codes. [ABSTRACT FROM AUTHOR]

Construction of maximal entanglement-assisted quantum error correction (EAQEC) codes is one of the fundamental problems of quantum computing and quantum information. The objective of this paper is twofold: firstly, to obtain all possible construction matrices of the linear codes over the skew group ring F 4 ⋊ φ G , where G is the cyclic and dihedral groups of finite orders; and secondly, to obtain some good maximal EAQEC codes over the finite field F 4 by using skew construction matrices. Additionally, to speed up the computational search time, we employ a nature inspired heuristic optimisation algorithm, the virus optimisation (VO) algorithm. With our method, we obtain a number of good maximal EAQEC codes over the finite field F 4 in a reasonably short time. In particular, we improve the lower bounds of 18 maximal EAQEC codes that exist in the literature. Moreover, some of our EAQEC codes turn out to be also maximum distance separable (MDS) codes. Also, by using our construction matrices, we provide counterexamples to Theorems 4 and 5 of Lai et al. (Quantum Inf Process 13(4):957–990, 2014), on the non-existence of maximal EAQEC codes with parameters [ [ n , 1 , n ; n - 1 ] ] and [ [ n , n - 1 , 2 ; 1 ] ] for an even length n. We also give a counterexample to another Theorem found in Lai and Ashikhmin (IEEE Trans Inf Theory 64:(1), 622–639, 2018), which states that there is no entanglement-assisted stabilizer code with parameters [ [ 4 , 2 , 3 ; 2 ] ] 4. [ABSTRACT FROM AUTHOR]

In this paper, we show that one can construct a $ G $-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which $ G $-codes are reversible of index $ \alpha $. Additionally, we introduce a new family of rings, $ {\mathcal{F}}_{j,k} $, whose base is the finite field of order $ 4 $ and study reversible $ G $-codes over this family of rings. Moreover, we present some possible applications of reversible $ G $-codes over $ {\mathcal{F}}_{j,k} $ to reversible DNA codes. We construct many reversible $ G $-codes over $ {\mathbb{F}}_4 $ of which some are optimal. These codes can be used to obtain reversible DNA codes. [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]

Let R be a commutative Noetherian ring with non-zero identity, 픞 an ideal of R and M a finitely generated R-module. We assume that N is a weakly Laskerian R-module and r is a non-negative integer such that the generalized local cohomology module is weakly Laskerian for all i < r. Then we prove that is also weakly Laskerian and so is finite. Moreover, we show that if s is a non-negative integer such that is weakly Laskerian for all i, j ≥ 0 with i ≤ s, then is weakly Laskerian for all i ≤ s and j ≥ 0. Also, over a Gorenstein local ring R of finite Krull dimension, we study the question when the socle of is weakly Laskerian? [ABSTRACT FROM AUTHOR]