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Characterizing a Generator Polynomial Matrix for the Dual of a multi-twisted code

MATHEMATICAL NOTES 2024年第3-4期116卷 777-792页

作者： Taki Eldin, R. F. Ain Shams Univ Fac Engn Cairo 11517 Egypt Egypt Univ Informat Cairo 15702 Egypt

The class of multi-twisted (MT) codes generalizes the classes of cyclic, constacyclic, quasi-cyclic, quasi-twisted, and generalized quasi-cyclic codes. We establish the correspondence between MT codes over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{F}_q$$\end{document} of index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{F}_q[x]$$\end{document}-submodules of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left(\mathbb{F}_q[x]\right)<^>\ell$$\end{document}. Thus, a basis of an MT code exists and is used to build a generator polynomial matrix (GPM). We prove some GPM properties, for example, relationship to code dimension, the identical equation, Hermite normal form. Hence, we prove a GPM formula for the dual code of an MT code. Finally, we obtain the necessary and sufficient conditions for the self-orthogonality and self-duality of MT codes.

关键词： multi-twisted code Generator polynomial matrix Dual code Hermite normal form

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Two-sided Galois duals of multi-twisted codes

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2023年第4期69卷 3459-3487页

作者： Taki Eldin, Ramy Ain Shams Univ Fac Engn Cairo Egypt

Galois duals of multi-twisted (MT) codes are considered in this study. We describe a MT code C as a module over a principal ideal domain. Hence, C has a generator polynomial matrix (GPM) G that satisfies an identical equation. We prove a GPM formula for the Euclidean dual C? using the identical equation of the Hermite normal form of G. Next, we aim to replace the Euclidean dual with the Galois dual. The Galois inner product is an asymmetric form, so we distinguish between the right and left Galois duals. We show that the right and left Galois duals of a MT code are MT as well but with possibly different shift constants. Some interconnected identities for the right and left Galois duals of a linear code are established and we also introduce the two-sided Galois dual. We use a condition that makes the two-sided Galois dual of a MT code MT, then we describe its GPM. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. A necessary and sufficient condition is established for the equality of the right and left Galois duals of any linear code.

关键词： multi-twisted code Generator polynomial matrix Two-sided Galois dual Hermite normal form

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