Yazar "Korban, Adrian" seçeneğine göre listele

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    Additive skew G-codes over finite fields
    (Springer Link, 2023) Dougherty, S. T.; Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz
    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 F4. Such codes have a strong connection to quantum error correcting codes.
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    An application of a virus optimization algorithm to the problem of computing binary self-dual and LCD codes
    (American Institute of Mathematical Sciences (AIMS), 2024) Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz
    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.
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    An Application of the Virus Optimization Algorithm to the Problem of Finding Extremal Binary Self-Dual Codes
    (2021) Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz
    In 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 | τ3(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.
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    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, Deniz
    We present a generator matrix of the form [σ(v1) | σ(v2)], where v1 ϵ RG and v2 ϵ 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.
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    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.
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    DNA Codes From Reversible Group Codes By A Virus Optimisation Algorithm
    (2023) Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz
    In 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.
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    DNA codes from skew dihedral group ring
    (American Institute of Mathematical Sciences (AIMS), 2022) Dougherty, S. T.; Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz
    In 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.
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    Group matrix ring codes and constructions of self-dual codes
    (Springer Link, 2023) Dougherty, S. T.; Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz
    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) 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.
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    Maximal entanglement-assisted quantum error correction codes from the skew group ring F4 (sic)φ G by a heuristic search scheme
    (Springer Link, 2022) Şahinkaya, Serap; Korban, Adrian; Üstün, Deniz
    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 (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).
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    Mutation-Based Algebraic Artificial Bee Colony Algorithm for Computing the Distance of Linear Codes
    (Dergipark, 2022) Üstün, Deniz; Şahinkaya, Serap; Korban, Adrian
    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).
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    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, Serap
    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 kth-range neighbours and search for binary [72,36,12] self-dual codes. In particular, we present six generator matrices of the form [I36 | τ6(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 γ=11,13,14,15,17,19,20,21,22,23,25,26,28,29,30,31,32 in their weight enumerators.
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    New singly and doubly even binary [72,36,12] self-dual codes from M2(R)G - group matrix rings
    (Academic Press in Elsevier Science, 2021) Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz
    In this work, we present a number of generator matrices of the form [I-2n vertical bar tau(2)(v)], where I-2n is the 2n x 2n 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 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.
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    New type I binary [72,36, 12] self-dual codes from M6(F2) G-Group matrix rings by a hybrid search technique based on a neighbourhood-virus optimisation algorithm
    (American Institute of Mathematical Sciences (AIMS), 2022) Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz
    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 kth -range neighbours, and search for binary [72, 36, 12] self-dual codes. In particular, we present six generator matrices of the form [I-36 vertical bar tau(6) (v)], where I-36 is the 36 x 36 identity matrix, v is an element in the group matrix ring M-6(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 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.
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    New Type I Binary [72,36,12] Self-Dual Codes From Composite Matrices And R-1 Lifts
    (American Institute of Mathematical Sciences (AIMS), 2023) Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz
    In this work, we define three composite matrices derived from group rings. We employ these composite matrices to create generator matrices of the form [In | Ω(v)], where In is the identity matrix and Ω(v) is a composite matrix and search for binary self-dual codes with parameters [36, 18, 6 or 8]. We next lift these codes over the ring R1 = F2 + uF2 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].
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    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, Deniz
    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.
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    Reversible G-codes over the ring Fj,k with applications to DNA codes
    (American Institute of Mathematical Sciences (AIMS), 2023) Korban, Adrian; Şahinkaya, Serap; Üstün, Deniz; Cengellenmis, Yasemin; Dertli, Abdullah; Dougherty, Steven T.
    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, F-j,F-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 F-j,F-k to reversible DNA codes. We construct many reversible G-codes over F-4 of which some are optimal. These codes can be used to obtain reversible DNA codes.
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    Reversible Gk-codes with applications to DNA codes
    (Springer Link, 2022) Korban, Adrian; Şahinkaya, Serap; Ü
    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 is an element of N, over the finite commutative Frobenius ring R. We employ our construction method to obtain many DNA codes over F-4 that satisfy the Hamming distance, the reverse, the 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 DNA codes of lengths 48, 56, 60, 64 and 72 for some fixed values of the Hamming distance