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Öğe A novel genetic search scheme based on nature-inspired evolutionary algorithms for binary self-dual codes(American Institute of Mathematical Sciences (AIMS), 2022) Korban, Adrian; Şahinkaya, Serap; Üstün, DenizIn 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.Öğe A novel method for image encryption using time signature-dependent s-boxes based on latin squares and the playfair system of cryptography(Springer Link, 2024) Dougherty, S. T.; Şahinkaya, Serap; ÜThis paper presents an image encryption algorithm by using time signature-dependent S-Boxes, which are based on Latin squares, the Playfair system of cryptography, and functions that are inspired by the behavior of a Japanese ladder. The encryption algorithm includes four stages: the construction of the S-Box, the generation of keys, image diffusion, and image permutation. The public key is generated from the grey-scale values of the plaintext image and the time signature, and secret key consists of the time signature and two functions from F82 to F82. Permutation and diffusion stages of the encryption algorithm are based on a given S-Box. Moreover, a chaotic map is used in the permutation phase for an effective shuf-fling of pixel positions. The simulation results and security analyses show that the proposed encryption scheme is quite secure and it can resist various cyber attacks effectively.Öğe Additive complementary dual codes from group characters(Institute of Electrical and Electronics Engineers (IEEE), 2022) Dougherty, S. T.; Şahinkaya, Serap; Üstün, DenizAdditive codes have become an increasingly important topic in algebraic coding theory due to their applications in quantum error-correction and quantum computing. Linear Complementary Dual (LCD) codes play an important role for improving the security of information against certain attacks. Motivated by these facts, we define additive complementary dual codes (ACD for short) over a finite abelian group in terms of an arbitrary duality on the ambient space and examine their properties. We show that the best minimum weight of ACD codes is always greater than or equal to the best minimum weight of LCD codes of the same size and that this inequality is often strict. We give some matrix constructions for quaternary ACD codes from a given quaternary ACD code and also from a given binary self-orthogonal code. Moreover, we construct an algorithm to determine if a given quaternary additive code is an ACD code with respect to all possible symmetric dualities. We also determine the largest minimum distance of quaternary ACD codes for lengths n <= 10. The obtained codes are either optimal or near optimal according to Bierbrauer et al+. (2009).Öğe Additive skew G-codes over finite fields(Springer Link, 2023) Dougherty, S. T.; Korban, Adrian; Şahinkaya, Serap; Üstün, DenizWe 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 F4. Such codes have a strong connection to quantum error correcting codes.Öğe An Application of the Virus Optimization Algorithm to the Problem of Finding Extremal Binary Self-Dual Codes(American Institute of Mathematical Sciences, 2021) Korban, Adrian; Şahinkaya, Serap; Üstün, DenizIn this paper, a virus optimization algorithm, which is one of the metaheuristic optimization technique, is employed for the first time to the problem of finding extremal binary self-dual codes. We present a number of generator matrices of the form [I36 | ?3(v)], where I36 is the 36 × 36 identity matrix, v is an element in the group matrix ring M3(F2)G and G is a finite group of order 12, which we then employ together with the the virus optimization algorithm and the genetic algorithm to search for extremal binary self-dual codes of length 72. We obtain that the virus optimization algorithm finds more extremal binary self-dual codes than the genetic algorithm. Moreover, by employing the above mentioned constructions together with the virus optimization algorithm, we are able to 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.Öğe An S-Box construction from exponentiation in finite fields and its application in RGB color image encryption(Springer, 2024) Klobusicky, J.; Dougherty, S. T.; Şahinkaya, Serap; Üstün, DenizIn this study, the utilization of exponentiation in finite fields is investigated for the purpose of generating pseudo-random sequences which have a crucial role in cryptographic applications. More precisely, a novel method for generating pseudo-random sequences is proposed to construct an initial S-Box which is a key component in various encryption schemes. In addition to that, a shuffling algorithm that leverages the pseudo-random sequences is developed to enhance the effectiveness of the initial S-Box. The utilization of the proposed S-Box is applied to the RGB color images to showcase its performance and robustness in an image encryption scheme.Öğe Approaching the Minimum Distance Problem by Algebraic Swarm-Based Optimizations(Dergipark, 2021) Şahinkaya, Serap; Üstün, DenizIn 1948, Claude Shannon, published ”A Mathematical Theory of Communication,” a seminal paper, which was about reliable data transmission over noisy channels [12]. Efficient and reliable data transmission, which can be done by some error-control techniques, are one of the main interests of coding theory. Error detecting and correcting capability are very important feature of a code and it is determined by the minimum distance of the code. Computing the minimum distance of a linear code C of large length is a difficult problem in coding theory. In [14], Vardy showed that this computation is an NP-hard type. The problem of finding minimum distance is getting harder when the size of the code grows. Therefore, some meta-heuristic algorithms have been used to approach the problem. In most of the existing literature, genetic algorithms are used for the considered problem. As far as our knowledge, among the algorithms in the literature that are based on swarm intelligence, only the ant colony algorithm (ACO) was used for the minimum-weight codeword problem [4,5]. It is well known that there is no heuristic algorithm which can perform good enough to solve optimization problems, please see [13] for details. . Therefore, it is natural to try the other swarm-based optimization techniques for the considered problem. In this paper, bat algorithm (BA) and firefly algorithm (FA) are applied to the minimum distance problem by integrating the algebraic operator to the handled algorithms. Most of the papers in the literature uses codewords as a search space for the minimum distance problem. Recently, generator matrices were considered as a search space, which turned out to be a better approach than using the codewords as a search space, please see [1] for details. In this work, we also consider generator matrices as a search space. In coding theory, the BCH codes or BoseOCoChaudhuriOCoHocquenghemcodes form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field. Effectiveness of the presented algorithm is controlled by running the algorithm on BCH codes since they are the standard codes with known minimum distance values [3, 9]Öğe Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme(American Institute of Mathematical Sciences (AIMS), 2022) Dougherty, S. T.; Korban, Adrian; Şahinkaya, Serap; Üstün, DenizWe present a generator matrix of the form [?(v1) | ?(v2)], where v1 ? RG and v2 ? 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 ? n ? 61 which are new to the literature.Öğe Codes from the Skew Ring ... - ...(American Mathematical Society, 2023) Dougherty, S. T.; Şahinkaya, Serap; Üstün, DenizIn this work, we study codes generated by elements in the skew group matrix ring M k ( R ) ? ? G M_k(R)\rtimes _{\varphi }G , where R R is a finite commutative Frobenius ring, G G is an arbitrary finite group, and ? \varphi is a group homomorphism from G G to A u t ( M k ( R ) ) Aut(M_k(R)) . We then determine all possible group homomorphisms ? : G ? A u t ( M 2 ( F 2 ) ) , \varphi : G \rightarrow Aut(M_2(\mathbb {F}_2)), for the cases where G G is a cyclic group and a dihedral group. Finally, by using skew generator matrices we provide examples of binary self-dual codes and also binary linear optimal codes.Öğe Construction of DNA Codes From Composite Matrices and a Bio-Inspired Optimization Algorithm(IEEE-INST Electrical Electronics Engineers Inc., 2023) Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz; Dougherty, S. T.Indexed keywords SciVal Topics Metrics Funding details Abstract In this work, we present a new construction method for reversible codes. We employ composite matrices derived from group rings and show how to construct these matrices so that they are also reversible. Also in this work, we give an algorithm for calculating conflict free DNA codes that satisfy the Hamming distance, the reverse, the reverse-complement, the GC-content constraints with each DNA codeword being free from reverse complement sub-strings. By employing our construction method for reversible codes and our algorithm, we construct a number of DNA codes that satisfy the above constraints. Many of the codes we obtain have better parameters than some known DNA codes and many have parameters that are new to the literature.Öğe Developing a secure image encryption technique using a novel S-box constructed through real-coded genetic algorithm’s crossover and mutation operators(Pergamon, 2024) Üstün, Deniz; Şahinkaya, Serap; Atlı, NurdanThe objective of this study is to craft a novel S-Box tailored to stringent security standards, achieved through iterative application of crossover and mutation operators inherent to real-coded genetic algorithms, ensuring robust image encryption. The designed S-Box is rigorously evaluated across multiple criteria, demonstrating its superiority over comparable S-Box designs found in existing literature. Furthermore, a secure image encryption method based on the designed S-Boxes devised and also including a 2D hyperchaotic Styblinski–Tang map. Thorough security assessments, encompassing statistical analysis and resilience testing against diverse attacks and noise, it is validated the encryption technique’s efficiency and applicability across varied scenarios.Öğe Dihedral codes with 1-dimensional hulls and 1-dimensional linear complementary pairs of dihedral codes(Springer Link, 2023) Dougherty, S. T.; Şahinkaya, Serap; Üstün, DenizIn this paper, we study dihedral codes with 1-dimensional hulls and we determine precisely when dihedral codes over fnite felds with 1-dimensional hulls exist. Moreover, we show that these codes come canonically in pairs. We also introduce 1-dimensional linear complementary pairs of dihedral codes and examine the properties of this class of codes. As an application, we obtain 1-dimensional linear complementary pair of dihedral codes, which are either optimal or near optimal.Öğe DNA Codes From Reversible Group Codes By A Virus Optimisation Algorithm(2023) Korban, Adrian; Şahinkaya, Serap; Üstün, DenizIn this paper, we employ group rings and some known results on group codes to study reversible group DNA codes. We define and study reversible cyclic DNA codes from a group ring point of view and we also introduce the notion for self-reciprocal group ring elements. Moreover, we search for reversible group DNA codes with the use of a virus optimisation algorithm. We obtain many good DNA codes that satisfy the Hamming distance, the reverse, the reverse-complement and the fixed GC-content constraints.Öğe DNA codes from skew dihedral group ring(American Institute of Mathematical Sciences (AIMS), 2022) Dougherty, S. T.; Korban, Adrian; Şahinkaya, Serap; Üstün, DenizIn this work, we present a matrix construction for reversible codes derived from skew dihedral group rings. By employing this matrix construction, the ring Fj,k and its associated Gray maps, we show how one can construct reversible codes of length n(2j+k )over the finite field 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.Öğe Dualities over the cross product of the cyclic groups of order 2(American Institute of Mathematical Sciences, 2024) Dougherty, S.T.; Şahinkaya, SerapWe 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 F2s. 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 |CÖğe Group matrix ring codes and constructions of self-dual codes(Springer Link, 2023) Dougherty, S. T.; Korban, Adrian; Şahinkaya, Serap; Üstün, DenizIn 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) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring M-k(R) are one sided ideals in the group matrix ring M-k(R)G and the corresponding codes over the ring R are 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.Öğe Kernel stable and uniquely generated modules(Publishing House of the Romanian Academy, 2021) Şahinkaya, Serap; Quynh, Truong CongA module theoretic notion of annihilator-stable rings is defined and some characterizations of it are studied. A module M is called kernel-stable if every element α ∈ End(M) satisfies the following condition: if α(M) + Kerβ = M, β ∈ End(M), then (α − γ)(m) ∈Kerβ for an automorphism γ of M and for all m ∈ M. For a pseudo-semi-projective module M, this notion is equivalent to that of uniquely generated module. © 2021, Publishing House of the Romanian Academy. All rights reserved.Öğe Maximal entanglement-assisted quantum error correction codes from the skew group ring F4 (sic)? G by a heuristic search scheme(Springer Link, 2022) Şahinkaya, Serap; Korban, Adrian; Üstün, DenizConstruction 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 (sic)(phi) 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 [En, 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).Öğe Mutation-Based Algebraic Artificial Bee Colony Algorithm for Computing the Distance of Linear Codes(Dergipark, 2022) Üstün, Deniz; Şahinkaya, Serap; Korban, AdrianFinding 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).Öğe New Extremal Binary Self-Dual Codes of Length 72 from M6(F2)G - Group Matrix Rings by a Hybrid Search Technique Based on a Neighbourhood-Virus Optimisation Algorithm(2021) Korban, Adrian; Üstün, Deniz; Şahinkaya, SerapIn 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 kth-range neighbours and search for binary [72,36,12] self-dual codes. In particular, we present six generator matrices of the form [I36 | ?6(v)], where I36 is the 36×36 identity matrix, v is an element in the group matrix ring M6(F2)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 F2. We construct 1471 new Type I binary [72,36,12] self-dual codes with the rare parameters ?=11,13,14,15,17,19,20,21,22,23,25,26,28,29,30,31,32 in their weight enumerators.
