PIR Schemes from nD-Cyclic and nD-Constacyclic Codes and Their Monomial Equivalence Markus Grassl, Burcu Ecem Karakaş, Ferruh Özbudak, Buket Özkaya Computational and Applied Mathematics, 2026 Private Information Retrieval (PIR) scheme aims to retrieve data from a database without revealing any details about the identity of the data. The PIR scheme for coded storage systems with colluding servers gives a better PIR rate when the storage code and retrieval code have transitive automorphism groups. In this work, we study the transitivity of n D-cyclic codes and then PIR schemes from them together with several examples of n D-cyclic codes with better PIR rates. Then, we show the monomial equivalence between n D-cyclic codes and certain n D-constacyclic codes, which can be used as an alternative family of transitive codes.
On some permutation trinomials in characteristic three Burcu Gülmez Temür, Buket Özkaya Hacettepe Journal of Mathematics and Statistics, 2025 In this paper, we determine the permutation properties of the polynomial $x^3+x^{q+2}-x^{4q-1}$ over the finite field $\\mathbb{F}_{q^2}$ in characteristic three. Moreover, we consider the trinomials of the form $x^{4q-1}+x^{2q+1} \\pm x^{3}$. In particular, we first show that $x^3+x^{q+2}-x^{4q-1}$ permutes $\\mathbb{F}_{q^2}$ with $q=3^m$ if and only if $m$ is odd. This enables us to show that the sufficient condition in [34, Theorem 4] is also necessary. Next, we prove that $x^{4q-1}+x^{2q+1} - x^{3}$ permutes $\\mathbb{F}_{q^2}$ with $q=3^m$ if and only if $m\\not\\equiv 0 \\pmod 4$. Consequently, we prove that the sufficient condition in [20, Theorem 3.2] is also necessary. Finally, we investigate the trinomial $x^{4q-1}+x^{2q+1} + x^{3}$ and show that it is never a permutation polynomial of $\\mathbb{F}_{q^2}$ in any characteristic. All the polynomials considered in this work are not quasi-multiplicative equivalent to any known class of permutation trinomials.
Optimum Distance Cyclic H-Orbit Full Flag Codes Zeynelabidin Karakaş, Ferruh Özbudak, Buket Özkaya IEEE Access, 2025 Flag codes, introduced as a special class of nested subspace codes, have recently attracted significant attention in the research community. In this work, we present constructions of optimum-distance cyclic <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i>-orbit full flag codes, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> is a cyclic subgroup of F<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∗</sup><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q<sup>n</sup></sub> of order at most <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i>/2</sup> + 1. Our aim is to extend certain results from [10] to the framework of flag codes. The proposed constructions, unlike the work [4], achieve full flags with optimum distance and either maximal or best-possible size, depending on the parity of the base field. We also provide explicit examples that demonstrate the structure of our codes. Furthermore, for the case of even base field size, we prove -via intricate arithmetical arguments- that our construction is unique under certain conditions.
Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes Martianus Frederic Ezerman, Markus Grassl, San Ling, Ferruh Özbudak, Buket Özkaya IEEE Transactions on Information Theory, 2025 Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database.
New distance bounds for quasi-cyclic codes Ferruh Özbudak, Buket Özkaya Designs Codes and Cryptography, 2024 We consider the minimum weight of codewords in a quasi-cyclic code and characterize the estimate in its most general setup using their concatenated structure. The new bound we derive generalizes the Jensen and Güneri–Özbudak bounds and it holds for the more general class of multilevel concatenated codes.
Complete characterization of a class of permutation trinomials in characteristic five Markus Grassl, Ferruh Özbudak, Buket Özkaya, Burcu Gülmez Temür Cryptography and Communications, 2024 In this paper, we address an open problem posed by Bai and Xia in [2]. We study polynomials of the form $$f(x)=x^{4q+1}+\\lambda _1x^{5q}+\\lambda _2x^{q+4}$$ f ( x ) = x 4 q + 1 + λ 1 x 5 q + λ 2 x q + 4 over the finite field $${\\mathbb F}_{5^{k}}$$ F 5 k , which are not quasi-multiplicative equivalent to any of the known permutation polynomials in the literature. We find necessary and sufficient conditions on $$\\lambda _1, \\lambda _2 \\in {\\mathbb F}_{5^{k}}$$ λ 1 , λ 2 ∈ F 5 k so that f(x) is a permutation monomial, binomial, or trinomial of $${\\mathbb F}_{5^{2k}}$$ F 5 2 k .
Improved Spectral Bound for Quasi-Cyclic Codes Gaojun Luo, Martianus Frederic Ezerman, San Ling, Buket Özkaya IEEE Transactions on Information Theory, 2024 Spectral bounds form a powerful tool to estimate the minimum distances of quasi-cyclic codes. They generalize the defining set bounds of cyclic codes to those of quasi-cyclic codes. Based on the eigenvalues of quasi-cyclic codes and the corresponding eigenspaces, we provide an improved spectral bound for quasi-cyclic codes. Numerical results verify that the improved bound outperforms the Jensen bound in almost all cases. Based on the improved bound, we propose a general construction of quasi-cyclic codes with excellent designed minimum distances. For the quasi-cyclic codes produced by this general construction, the improved spectral bound is always sharper than the Jensen bound.
QUASI-TWISTED CODES AS CONTRACTIONS OF QUASI-CYCLIC CODES Ferruh Özbudak, Buket Özkaya Advances in Mathematics of Communications, 2024 We consider the quasi-twisted codes as contractions of quasi-cyclic codes and construct a family of $ q $-ary quasi-cyclic codes whose codewords have $ r $-divisible weights, where $ r\mid q-1 $. We show that any quasi-cyclic code of co-index divisible by $ r $ is a direct sum of $ r $-divisible quasi-cyclic codes.
Spectral Bounds for Quasi-Twisted Codes Martianus Frederic Ezerman, San Ling, Buket Ozkaya, Jareena Tharnnukhroh IEEE International Symposium on Information Theory Proceedings, 2019
On Linear Complementary Pairs of Codes Claude Carlet, Cem Guneri, Ferruh Ozbudak, Buket Ozkaya, Patrick Sole IEEE Transactions on Information Theory, 2018
PIR Schemes from n D-Cyclic and n D-Constacyclic Codes and Their Monomial Equivalence: M. Grassl et al. M Grassl, BE Karakaş, F Özbudak, B Özkaya Computational and Applied Mathematics 45 (6), 246 , 2026 2026
Extended LP bound for LCD codes and new binary and ternary LCD codes E Karabakla, B Özkaya Applicable Algebra in Engineering, Communication and Computing, 1-37 , 2026 2026
Generalized Spectral Bound for Quasi-Twisted Codes B Özkaya arXiv preprint arXiv:2511.10066 , 2025 2025
Optimum Distance Cyclic H-Orbit Full Flag Codes Z Karakaş, F Özbudak, B Özkaya IEEE Access 13, 194149-194155 , 2025 2025
New distance bounds for quasi-cyclic codes F Özbudak, B Özkaya Designs, Codes and Cryptography 92 (12), 3981-4009 , 2024 2024 Citations: 2
Characterization of nearly self-orthogonal quasi-twisted codes and related quantum codes MF Ezerman, M Grassl, S Ling, F Özbudak, B Özkaya IEEE Transactions on Information Theory 71 (1), 499-517 , 2024 2024 Citations: 12
On a class of permutation trinomials over finite fields B Özkaya, B Gülmez Temür 2024
On some permutation trinomials in characteristic three B Özkaya, B Gülmez Temür 2024
New distance bounds for quasi-cyclic codes B Özkaya, F Özbudak 2024
Complete characterization of a class of permutation trinomials in characteristic five M Grassl, F Özbudak, B Özkaya, BG Temür Cryptography and Communications 16 (4), 825-841 , 2024 2024 Citations: 3
Quasi-twisted codes as contractions of quasi-cyclic codes. F Özbudak, B Özkaya Adv. Math. Commun. 18 (2), 394-409 , 2024 2024
Improved spectral bound for quasi-cyclic codes G Luo, MF Ezerman, S Ling, B Özkaya IEEE Transactions on Information Theory 70 (6), 4002-4015 , 2024 2024 Citations: 7
On some permutation trinomials in characteristic three BG Temür, B Özkaya Hacettepe Journal of Mathematics and Statistics, 1-10 , 2024 2024 Citations: 1
On a class of permutation trinomials over finite fields B GÜLMEZ TEMÜR, B ÖZKAYA Turkish Journal of Mathematics 48 (4), 778-792 , 2024 2024
Quasi-twisted codes as contractions of quasi-cyclic codes B Özkaya, F Özbudak 2023
A comparison of distance bounds for quasi-twisted codes MF Ezerman, JM Lampos, S Ling, B Özkaya, J Tharnnukhroh IEEE Transactions on Information Theory 67 (10), 6476-6490 , 2021 2021 Citations: 10
Quasi-cyclic codes C Güneri, S Ling, B Özkaya Concise Encyclopedia of Coding Theory, 129-150 , 2021 2021 Citations: 9
Quasi-cyclic codes B Özkaya, C Güneri, S Ling A Concise Encyclopedia of Coding Theory , 2021 2021
New bounds on the minimum distance of cyclic codes S Ling, B Özkaya Advances in Mathematics of Communications 15 (1), 1-8 , 2021 2021
Buket Özkaya B Özkaya 2020
MOST CITED SCHOLAR PUBLICATIONS
The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices ST Dougherty, JL Kim, B Ozkaya, L Sok, P Solé Int. J. Information and Coding Theory, 4 (2/3), 116-128 , 2017 2017 Citations: 177
Quasi-cyclic complementary dual codes C Güneri, B Özkaya, P Solé Finite Fields and Their Applications 42, 67-80 , 2016 2016 Citations: 131
On linear complementary pairs of codes C Carlet, C Güneri, F Özbudak, B Özkaya, P Sole IEEE Transactions on Information Theory 64 (10), 6583-6589 , 2018 2018 Citations: 90
Structure and performance of generalized quasi-cyclic codes C Güneri, F Özbudak, B Özkaya, E Saçıkara, Z Sepasdar, P Sole Finite Fields and Their Applications 47, 183-202 , 2017 2017 Citations: 55
Good Stabilizer Codes from Quasi-Cyclic Codes over F 4 and F 9 MF Ezerman, S Ling, B Özkaya, P Solé 2019 IEEE international symposium on information theory (ISIT), 2898-2902 , 2019 2019 Citations: 40
On self-dual double negacirculant codes A Alahmadi, C Güneri, B Özkaya, H Shoaib, P Solé Discrete Applied Mathematics 222, 205-212 , 2017 2017 Citations: 34
On Linear Complementary Pair of D Cyclic Codes C Güneri, B Özkaya, S Sayıcı IEEE Communications Letters 22 (12), 2404-2406 , 2018 2018 Citations: 29
On complementary dual multinegacirculant codes A Alahmadi, C Güneri, B Özkaya, H Shoaib, P Solé Cryptography and Communications, 1-13 , 2019 2019 Citations: 16
Multidimensional quasi-cyclic and convolutional codes C Güneri, B Özkaya IEEE Transactions on Information Theory 62 (12), 6772-6785 , 2016 2016 Citations: 13
Characterization of nearly self-orthogonal quasi-twisted codes and related quantum codes MF Ezerman, M Grassl, S Ling, F Özbudak, B Özkaya IEEE Transactions on Information Theory 71 (1), 499-517 , 2024 2024 Citations: 12
A comparison of distance bounds for quasi-twisted codes MF Ezerman, JM Lampos, S Ling, B Özkaya, J Tharnnukhroh IEEE Transactions on Information Theory 67 (10), 6476-6490 , 2021 2021 Citations: 10
A minimum distance bound for quasi-nD-cyclic codes F Özbudak, B Özkaya Finite Fields and their Applications 41, 193-222 , 2016 2016 Citations: 10
Quasi-cyclic codes C Güneri, S Ling, B Özkaya Concise Encyclopedia of Coding Theory, 129-150 , 2021 2021 Citations: 9
Multidimensional quasi-twisted codes: equivalent characterizations and their relation to multidimensional convolutional codes S Ling, B Özkaya Designs, Codes and Cryptography 87 (12), 2941-2965 , 2019 2019 Citations: 8
Spectral Bounds for Quasi-Twisted Codes MF Ezerman, S Ling, B Özkaya, J Tharnnukhroh 2019 IEEE International Symposium on Information Theory (ISIT), 1922-1926 , 2019 2019 Citations: 8
Improved spectral bound for quasi-cyclic codes G Luo, MF Ezerman, S Ling, B Özkaya IEEE Transactions on Information Theory 70 (6), 4002-4015 , 2024 2024 Citations: 7
Multidimensional quasi-cyclic and convolutional codes B Özkaya 2014 Citations: 4
Complete characterization of a class of permutation trinomials in characteristic five M Grassl, F Özbudak, B Özkaya, BG Temür Cryptography and Communications 16 (4), 825-841 , 2024 2024 Citations: 3
New distance bounds for quasi-cyclic codes F Özbudak, B Özkaya Designs, Codes and Cryptography 92 (12), 3981-4009 , 2024 2024 Citations: 2
On some permutation trinomials in characteristic three BG Temür, B Özkaya Hacettepe Journal of Mathematics and Statistics, 1-10 , 2024 2024 Citations: 1
