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Characterization of Nearly Self-Orthogonal quasi-twisted codes and Related Quantum codes

IEEE TRANSACTIONS ON INFORMATION THEORY 2025年第1期71卷 499-517页

作者： Ezerman, Martianus Frederic Grassl, Markus Ling, San Ozbudak, Ferruh Ozkaya, Buket Nanyang Technol Univ Sch Phys & Math Sci Singapore 637371 Singapore PQStat Singapore 408564 Singapore Univ Gdansk Int Ctr Theory Quantum Technol PL-80309 Gdansk Poland Sabanci Univ Fac Engn & Nat Sci TR-34956 Istanbul Turkiye Middle East Tech Univ Inst Appl Math TR-06800 Ankara Turkiye

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.

关键词： codes Quantum mechanics Additives Vectors Linear codes Galois fields Tensors Qubit Indexes Hamming weight Construction X Hermitian hull quantum stabilizer code quasi-twisted code

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Multidimensional quasi-twisted codes: equivalent characterizations and their relation to multidimensional convolutional codes

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DESIGNS codeS AND CRYPTOGRAPHY 2019年第12期87卷 2941-2965页

作者： Ling, San Ozkaya, Buket Nanyang Technol Univ Sch Phys & Math Sci Div Math Sci Singapore 637371 Singapore

We study multidimensional analogues of quasi-twisted codes from different points of view. Their concatenated structure allows us to characterize self-dual and complementary-dual classes of such codes as well as to show that multidimensional quasi-twisted (QT) codes are asymptotically good, together with their self-dual and complementary-dual subclasses. They are naturally related to nD convolutional codes as well. It is known that the minimum distance of quasi-cyclic codes provides a lower bound on the free distance of convolutional codes. An analogous result was shown for certain 1-generator 2D convolutional codes by using quasi-2D-cyclic codes. We prove a similar relation between convolutional codes and the related QT codes first, and then generalize the relation further to certain product convolutional codes and the related product QT codes, which improves the previous result in terms of dimension and number of generators. We also provide two-dimensional ternary and binary codes of modest lengths which yield good parameters.

关键词： quasi-twisted code Constacyclic code Abelian code Convolutional code

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On quasi-twisted codes over finite fields

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FINITE FIELDS AND THEIR APPLICATIONS 2012年第2期18卷 237-257页

作者： Jia, Yan Nanyang Technol Univ SPMS MAS 0301 Div Math Sci Singapore 637371 Singapore

In coding theory, quasi-twisted (QT) codes form an important class of codes which has been extensively studied. We decompose a QT code to a direct sum of component codes linear codes over rings. Furthermore, given the decomposition of a QT code, we can describe the decomposition of its dual code. We also use the generalized discrete Fourier transform to give the inverse formula for both the nonrepeated-root and repeated-root cases. Then we produce a formula which can be used to construct a QT code given the component codes. (C) 2011 Elsevier Inc. All rights reserved.

关键词： quasi-twisted code Finite field Generalized discrete Fourier transform Repeated-root case Nonrepeated-root case Construction

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A Comparison of Distance Bounds for quasi-twisted codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2021年第10期67卷 6476-6490页

作者： Ezerman, Martianus Frederic Lampos, John Mark Ling, San Ozkaya, Buket Tharnnukhroh, Jareena Nanyang Technol Univ Sch Phys & Math Sci Singapore 637371 Singapore Univ Philippines Inst Math Sci & Phys Los Banos 4031 Laguna Philippines

Spectral bounds on the minimum distance of quasi-twisted codes over finite fields are proposed, based on eigenvalues of polynomial matrices and the corresponding eigenspaces. They generalize the Semenov-Trifonov and Zeh-Ling bounds in a way similar to how the Roos and shift bounds extend the BCH and HT bounds for cyclic codes. The eigencodes of a quasi-twisted code in the spectral theory and the outer codes in its concatenated structure are related. A comparison based on this relation verifies that the Jensen bound always outperforms the spectral bound under special conditions, which yields a similar relation between the Lally and the spectral bounds. The performances of the Lally, Jensen and spectral bounds are presented in comparison with each other.

关键词： Eigenvalues and eigenfunctions Manganese Spectral analysis Scholarships Indexes Generators Product codes quasi-twisted code concatenated code minimum distance bound polynomial matrices spectral analysis

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Matrix product and quasi-twisted codes in one class

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第2期43卷 1-20页

作者： Eldin, Ramy Taki Ain Shams Univ Fac Engn Cairo Egypt Egypt Univ Informat Knowledge City New Adm Capital Egypt

Many classical constructions, such as Plotkin's and Turyn's, were generalized by matrix product (MP) codes. quasi-twisted (QT) codes, on the other hand, form an algebraically rich structure class that contains many codes with best-known parameters. We significantly extend the definition of MP codes to establish a broader class of generalized matrix product (GMP) codes that contains QT codes as well. We propose a generator matrix formula for any linear GMP code and provide a condition for determining the code size. We prove that any QT code has a GMP structure. Then we show how to build a generator polynomial matrix for a QT code from its GMP structure, and vice versa. Even though the class of QT codes contains many codes with best-known parameters, we present different examples of GMP codes with best-known parameters that are neither MP nor QT. Two different lower bounds on the minimum distance of GMP codes are presented;they generalize their counterparts in the MP codes literature. The second proposed lower bound replaces the non-singular by columns matrix with a less restrictive condition. Some examples are provided for comparing the two proposed bounds, as well as showing that these bounds are tight.

关键词： Matrix product code quasi-twisted code Best-known parameter Lower bound

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Spectral Bounds for quasi-twisted codes

Spectral Bounds for Quasi-Twisted Codes

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IEEE International Symposium on Information Theory (ISIT)

作者： Ezerman, Martianus Frederic Ling, San Ozkaya, Buket Tharnnukhroh, Jareena Nanyang Technol Univ Sch Phys & Math Sci 21 Nanyang Link Singapore 637371 Singapore

New lower bounds on the minimum distance of quasi-twisted codes over finite fields are proposed. They are based on spectral analysis and eigenvalues of polynomial matrices. They generalize the Semenov-Trifonov and Zeh... 详细信息

ISBN: (纸本)9781538692912

关键词： quasi-twisted code Roos bound shift bound eigenvalues polynomial matrices spectral analysis

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A class of 1-generator right quasi-polycyclic codes 2

A class of 1-generator right Quasi-polycyclic Codes

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2nd International Conference on Innovative Research in Applied Science, Engineering and Technology (IRASET)

作者： Ou-Azzou, Hassan Ialou, Brahim Mouloua, El Mahdi Moulay Ismail Univ Dept Math ENSAM ENSAM Meknes Morocco

ISBN: (纸本)9781665422093

In the present paper, we interest in right l-quasipolycyclic (((a) over right arrow, l)-QP) codes of length n = ml induced by the vector (a) over right arrow = (a(0), a(1),..., a(m-1)) is an element of F-q(m), over the finite field F-q, which forms a generalization of quasi-cyclic, quasi-negacyclic and quasi-twisted codes. After recalling the necessary background on polycyclic and QP codes, we discuss the structure and the enumeration of the 1-generator l-QP codes. Finally, we give some examples as applications of some of the results using the SageMath software.

关键词： Polycyclic code quasi-cyclic code quasi-twisted code Invariant subspace

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A note on skew constacyclic codes over F_q

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DISCRETE MATHEMATICS ALGORITHMS AND APPLICATIONS 2019年第3期11卷

作者： Islam, Habibul Prakash, Om Indian Inst Technol Patna Dept Math Patna 801106 Bihar India

In this paper, the skew constacyclic codes over finite non-chain ring R = F-q + uF(q) + vF(q), where q = p(m), p is an odd prime and u(2) = u, v(2) = v, uv = vu = 0, are studied. We show that the Gray image of a skew alpha-constacyclic, code of length n over R. is a skew quasi-twisted code of length 3n over F-q of index 3. Further, the structural properties of skew constacyclic codes over R are obtained by the decomposition method.

关键词： Skew polynomial ring skew constacyclic code quasi-twisted code Gray map idempotent generator non-chain ring

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