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

Finding the minimum distance of linear codes is a non-deterministic polynomial-time-hard problem and different approaches are used in the literature to solve this problem. Although, some of the methods focus on finding the true distances by using exact algorithms, some of them focus on optimization algorithms to find the lower or upper bounds of the distance. In this study, we focus on the latter approach. We first give the swarm intelligence background of artificial bee colony algorithm, we explain the algebraic approach of such algorithm and call it the algebraic artificial bee colony algorithm (A-ABC). Moreover, we develop the A-ABC algorithm by integrating it with the algebraic differential mutation operator. We call the developed algorithm the mutation-based algebraic artificial bee colony algorithm (MBA-ABC). We apply both; the A-ABC and MBA-ABC algorithms to the problem of finding the minimum distance of linear codes. The achieved results indicate that the MBA-ABC algorithm has a superior performance when compared with the A-ABC algorithm when finding the minimum distance of Bose, Chaudhuri, and Hocquenghem (BCH) codes (a special type of linear codes).

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 $k, n \in \mathbb{N},$ over the finite commutative Frobenius ring $R.$ We employ our construction method to obtain many DNA codes over $\mathbb{F}_4$ that satisfy the Hamming distance, reverse, 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 some DNA codes of lengths $48, 56, 60, 64$ and $72$ for some fixed values of the Hamming distance $d.$

In this paper, we extend the work done on $G$-codes over formal power series rings and finite chain rings $\mathbb{F}_q[t]/(t^i)$, to composite $G$-codes over the same alphabets. We define composite $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G.$ 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 $\gamma$-adic $G$-codes over $R_\infty$ to composite $G$-codes and study these codes over principal ideal rings.

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 $${\mathbb {F}}_4$$ . Such codes have a strong connection to quantum error correcting codes.

In this work, we study a new family of rings, ${\mathscr{B}}_{j,k}$ B j , k , whose base field is the finite field ${\mathbb {F}}_{p^{r}}$ F p r . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over ${\mathscr{B}}_{j,k}$ B j , k to a code over ${\mathscr{B}}_{l,m}$ B l , m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible $G^{2^{j+k}}$ G 2 j + k -code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s.

In this work, we extend an established isomorphism between group rings and a subring of the [Formula: see text] matrices. This extension allows us to construct more complex matrices over the ring [Formula: see text] 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.

We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group $${\mathbb {Z}}_{p^e}$$ . They exist for all lengths when p is prime and e is even; all even lengths when p is an odd prime with $$p \equiv 1 \pmod {4}$$ and e is odd with $$e>1$$ ; and all lengths that are $$0 \pmod {4}$$ when p is an odd prime with $$p \equiv 3 \pmod {4}$$ and e is odd with $$e>1.$$

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 \begin{document}$ \mathbb{F}_4 $\end{document} . These codes have binary images with parameters \begin{document}$ [32,16,8] $\end{document} or \begin{document}$ [32,16,6] $\end{document} . These are lifted to codes over \begin{document}$ \mathbb{F}_4+u\mathbb{F}_4 $\end{document} , to obtain codes with Gray images of extremal self-dual binary codes of length 64. Finally, we use a building-up method over \begin{document}$ \mathbb{F}_2+u\mathbb{F}_2 $\end{document} 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.

In this work, we present a matrix construction for reversible codes derived from skew dihedral group rings. By employing this matrix construction, the ring \begin{document}$ \mathcal{F}_{j, k} $\end{document} and its associated Gray maps, we show how one can construct reversible codes of length \begin{document}$ n2^{j+k} $\end{document} over the finite field \begin{document}$ \mathbb{F}_4. $\end{document} 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.

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.

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 \begin{document}$ k^{th} $\end{document}-range neighbours, and search for binary \begin{document}$ [72, 36, 12] $\end{document} self-dual codes. In particular, we present six generator matrices of the form \begin{document}$ [I_{36} \ | \ \tau_6(v)], $\end{document} where \begin{document}$ I_{36} $\end{document} is the \begin{document}$ 36 \times 36 $\end{document} identity matrix, \begin{document}$ v $\end{document} is an element in the group matrix ring \begin{document}$ M_6(\mathbb{F}_2)G $\end{document} and \begin{document}$ G $\end{document} is a finite group of order 6, to which we employ the proposed algorithm and search for binary \begin{document}$ [72, 36, 12] $\end{document} self-dual codes directly over the finite field \begin{document}$ \mathbb{F}_2 $\end{document}. We construct 1471 new Type I binary \begin{document}$ [72, 36, 12] $\end{document} self-dual codes with the rare parameters \begin{document}$ \gamma = 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32 $\end{document} in their weight enumerators.

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

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 $n \times n$ 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., 17 pages

In this work, we apply the idea of composite matrices arising from group rings to derive a number of different techniques for constructing self-dual codes over finite commutative Frobenius rings. By applying these techniques over different alphabets, we construct best known singly-even binary self-dual codes of lengths 80, 84 and 96 as well as doubly-even binary self-dual codes of length 96 that were not known in the literature before., arXiv admin note: text overlap with arXiv:2102.10354

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.

In this paper, we extend the work done on $G$-codes over formal power series rings and finite chain rings $\mathbb{F}_q[t]/(t^i)$, to composite $G$-codes over the same alphabets. We define composite $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G.$ 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 $\gamma$-adic $G$-codes over $R_\infty$ to composite $G$-codes and study these codes over principal ideal rings.

In this work, we present a number of generator matrices of the form $[I_{2n} \ | \ \tau_k(v)],$ where $I_{kn}$ is the $kn \times kn$ 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 $\mathbb{F}_2.$ 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., Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:2102.00475; text overlap with arXiv:2102.00474

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 $$M_k(R)$$ M k ( R ) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring $$M_k(R)$$ M k ( R ) are one sided ideals in the group matrix ring $$M_k(R)G$$ M k ( R ) G and the corresponding codes over the ring R are $$G^k$$ 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.

In this paper, a new search technique based on the 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 is a known in the literature approach 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, which we then employ to 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., Comment: arXiv admin note: text overlap with arXiv:2103.07739, arXiv:2102.12863

In this work, we define $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. 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 $\gamma$-adic codes over $R_\infty$ to $\gamma$-adic $G$-codes over the same ring. We also study $G$-codes over principal ideal rings.

In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We define quasi-composite G-codes and give a construction of these codes. We also study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters gamma=7,8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions., Comment: 33 pages

Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form \begin{document}$ G = (I_n \ | \ A), $\end{document} where \begin{document}$ I_n $\end{document} is the \begin{document}$ n \times n $\end{document} identity matrix and \begin{document}$ A $\end{document} is the \begin{document}$ n \times n $\end{document} matrix fully determined by the first row. In this work, we define a generator matrix in which \begin{document}$ A $\end{document} is a block matrix, where the blocks come from group rings and also, \begin{document}$ A $\end{document} is not fully determined by the elements appearing in the first row. By applying our construction over \begin{document}$ \mathbb{F}_2+u\mathbb{F}_2 $\end{document} and by employing the extension method for codes, we were able to construct new extremal binary self-dual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to construct many new binary self-dual \begin{document}$ [68, 34, 12] $\end{document}-codes with the rare parameters \begin{document}$ \gamma = 7, 8 $\end{document} and \begin{document}$ 9 $\end{document} in \begin{document}$ W_{68, 2}. $\end{document} In particular, we find 92 new binary self-dual \begin{document}$ [68, 34, 12] $\end{document}-codes.

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.

In this work, we describe a construction in which we combine together the idea of a bisymmetric matrix and group rings. Applying this construction over the ring F 4 + u F 4 together with the well known extension and neighbour methods, we construct new self-dual codes of length 68 . In particular, we find 41 new codes of length 68 that were not known in the literature before.

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 .

We describe a composite construction from group rings where the groups have orders 16 and 8. This construction is then applied to find the extremal binary self-dual codes with parameters [32, 16, 8] or [32, 16, 6]. We also extend this composite construction by expanding the search field which enables us to find more extremal binary self-dual codes with the above parameters and with different orders of automorphism groups. These codes are then lifted to 𝔽2 + u𝔽2, to obtain extremal binary images of codes of length 64. Finally, we use the extension method and neighbour construction to obtain new extremal binary self-dual codes of length 68. As a result, we obtain 28 new codes of length 68 which were not known in the literature before.