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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, 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]

In this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary self-dual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92. [ABSTRACT FROM AUTHOR]

In this work, we extend an established isomorphism between group rings and a subring of the n × n matrices. This extension allows us to construct more complex matrices over the ring R. We present many interesting examples of complex matrices constructed directly from our extension. We also show that some of the matrices used in the literature before can be obtained by a direct application of our extended isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we extend the work done on G-codes over formal power series rings and finite chain rings Fq[t]=(ti), to composite G-codes over the same alphabets. We define composite G-codes over the infinite ring R1 as ideals in the group ring R1G: We show that the dual of a composite G-code is again a composite G-code in this setting. We extend the known results on projections and lifts of G-codes over the finite chain rings and over the formal power series rings to composite G-codes. Additionally, we extend some known results on -adic G-codes over R1 to composite G-codes and study these codes over principal ideal rings. [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]

We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over F4. These codes have binary images with parameters [32, 16, 8] or [32, 16, 6]. These are lifted to codes over F4 + uF4, to obtain codes with Gray images of extremal self-dual binary codes of length 64. Finally, we use a building-up method over F2 + uF2 to obtain new extremal binary self-dual codes of length 68. We construct 11 new codes via the building-up method and 2 new codes by considering possible neighbors. [ABSTRACT FROM AUTHOR]

In this work, we define G-codes over the infinite ring R∞ as ideals in the group ring R∞G. We show that the dual of a G-code is again a G-code in this setting. We study the projections and lifts of G-codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing -adic codes over R∞ to γ-adic G-codes over the same ring. We also study G-codes over principal ideal rings. [ABSTRACT FROM AUTHOR]

In this work, we give a new technique for constructing self-dual codes over commutative Frobenius rings using λ -circulant matrices. The new construction was derived as a modification of the well-known four circulant construction of self-dual codes. Applying this technique together with the building-up construction, we construct singly-even binary self-dual codes of lengths 56, 58, 64, 80 and 92 that were not known in the literature before. Singly-even self-dual codes of length 80 with β ∈ { 2 , 4 , 5 , 6 , 8 } in their weight enumerators are constructed for the first time in the literature. [ABSTRACT FROM AUTHOR]

Abstract We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring F 2 + u F 2 , using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper. Highlights • A novel bordered construction for self-dual codes can be obtained using groups rings. • Extremal binary self-dual codes of different lengths can be constructed using the constructions. • 41 new additions to the set of known extremal binary self-dual codes of length 68 have been made. • There is a strong connection between groups rings and constructions for self-dual codes. [ABSTRACT FROM AUTHOR]

In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the n × n matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain condition on the group ring element v , one can construct non-trivial group LCD codes. Moreover, we also show that by adding more constraints on the group ring element v , one can construct group LCD codes that are reversible. We present many examples of binary group LCD codes of which some are optimal and group reversible LCD codes with different parameters. [ABSTRACT FROM AUTHOR]

In this work, we use the concept of distance between self-dual codes, which generalizes the concept of a neighbor for self-dual codes. Using the k -neighbors, we are able to construct extremal binary self-dual codes of length 68 with new weight enumerators. We construct 143 extremal binary self-dual codes of length 68 with new weight enumerators including 42 codes with γ = 8 in their W 68 , 2 and 40 with γ = 9 in their W 68 , 2. These examples are the first in the literature for these γ values. This completes the theoretical list of possible values for γ in W 68 , 2. [ABSTRACT FROM AUTHOR]