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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]

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 define three composite matrices derived from group rings. We employ these composite matrices to create generator matrices of the form $ [I_n \ | \ \Omega(v)], $ where $ I_n $ is the identity matrix and $ \Omega(v) $ is a composite matrix and search for binary self-dual codes with parameters $ [36,18, 6 \ \text{or} \ 8]. $ We next lift these codes over the ring $ R_1 = \mathbb{F}_2+u\mathbb{F}_2 $ to obtain codes whose binary images are self-dual codes with parameters $ [72,36,12]. $ Many of these codes turn out to have weight enumerators with parameters that were not known in the literature before. In particular, we find $ 30 $ new Type I binary self-dual codes with parameters $ [72,36,12]. $ [ABSTRACT FROM AUTHOR]

In this paper, a new search technique based on a virus optimisation algorithm is proposed for calculating the neighbours of binary self-dual codes. The aim of this new technique is to calculate neighbours of self-dual codes without reducing the search field in the search process (this technique is known in the literature due to the computational time constraint) but still obtaining results in a reasonable time (significantly faster when compared to the standard linear computational search). We employ this new search algorithm to the well-known neighbour method and its extension, the $ k^{th} $-range neighbours, and search for binary $ [72, 36, 12] $ self-dual codes. In particular, we present six generator matrices of the form $ [I_{36} \ | \ \tau_6(v)], $ where $ I_{36} $ is the $ 36 \times 36 $ identity matrix, $ v $ is an element in the group matrix ring $ M_6(\mathbb{F}_2)G $ and $ G $ is a finite group of order 6, to which we employ the proposed algorithm and search for binary $ [72, 36, 12] $ self-dual codes directly over the finite field $ \mathbb{F}_2 $. We construct 1471 new Type I binary $ [72, 36, 12] $ self-dual codes with the rare parameters $ \gamma = 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32 $ in their weight enumerators. [ABSTRACT FROM AUTHOR]

Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form G = (I n | A) , G = (I n | A) , where I n I n is the n × n n × n identity matrix and A A is the n × n n × n matrix fully determined by the first row. In this work, we define a generator matrix in which A A is a block matrix, where the blocks come from group rings and also, A A is not fully determined by the elements appearing in the first row. By applying our construction over F 2 + u F 2 F 2 + u F 2 and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual [ 68 , 34 , 12 ] [ 68 , 34 , 12 ] -codes with the rare parameters γ = 7 , 8 γ = 7 , 8 and 9 9 in W 68 , 2. W 68 , 2. In particular, we find 92 new binary self-dual [ 68 , 34 , 12 ] [ 68 , 34 , 12 ] -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]