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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 present a generator matrix of the form $ [ \sigma(v_1) \ | \ \sigma(v_2)] $, where $ v_1 \in RG $ and $ v_2\in RH $, for finite groups $ G $ and $ H $ of order $ n $ for constructing self-dual codes and linear complementary dual codes over the finite Frobenius ring $ R $. In general, many of the constructions to produce self-dual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find self-dual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve $ 10 $ of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain $ 82 $ new binary linear complementary dual codes, $ 50 $ of which are either optimal or near optimal of lengths $ 41 \leq n \leq 61 $ which are new to the literature. [ABSTRACT FROM AUTHOR]

In this paper, a genetic algorithm, one of the evolutionary algorithm optimization methods, is used for the first time for the problem of computing extremal binary self-dual codes. We present a comparison of the computational times between the genetic algorithm and a linear search for different size search spaces and show that the genetic algorithm is capable of computing binary self-dual codes significantly faster than the linear search. Moreover, by employing a known matrix construction together with the genetic algorithm, we are able to obtain new binary self-dual codes of lengths 68 and 72 in a significantly short time. In particular, we obtain 11 new binary self-dual codes of length 68 and 17 new binary self-dual codes of length 72. [ABSTRACT FROM AUTHOR]

In this work, we define a modification of a bordered construction for self-dual codes which utilises $ \lambda $-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary self-dual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist. [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]