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MULTI-TWISTED CODES AS FREE MODULES OVER PRINCIPAL IDEAL DOMAINS.

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Název: MULTI-TWISTED CODES AS FREE MODULES OVER PRINCIPAL IDEAL DOMAINS.
Autoři: ELDIN, RAMY TAKI
Zdroj: Algebraic Structures & their Applications; Aug2025, Vol. 12 Issue 3, p207-241, 35p
Témata: LINEAR codes, CYCLIC codes, INTEGRAL domains, MATRICES (Mathematics), BINARY codes, PROJECTIVE modules (Algebra)
Abstract (English): We begin by introducing the simple algebraic structure of cyclic, constacyclic, quasi-cyclic (QC), quasi-twisted (QT), generalized quasi-cyclic (GQC), and multi-twisted (MT) codes over finite fields. Then, we establish the correspondence between these codes and submodules of the free Fq[x]-module (Fq[x])l. We show that an MT code is a linear code over the principal ideal domain (PID) Fq[x]. Hence, a basis of this code exists and is used to build a generator matrix with polynomial entries, called the generator polynomial matrix (GPM). The Hermite normal form of matrices over PIDs is exploited to achieve the reduced GPMs of MT codes. Some properties of the reduced GPM are introduced, for example, the identical equation. A formula for a GPM of the dual code of an MT code is established. At this point, special attention is paid to QC codes. We characterize GPMs for QC codes that combine reversibility and self-duality/self-orthogonality. We show the existence of binary self-orthogonal reversible QC codes that have the best known parameters as linear codes. [ABSTRACT FROM AUTHOR]
Abstract (Arabic): المقال يركز على دراسة الأكواد متعددة الالتواء (MT) كعناصر حرة فوق مجالات المثالي الرئيسي (PIDs)، موضحًا هياكلها الجبرية وتطبيقاتها في تصحيح الأخطاء. يقدم أنواعًا مختلفة من الأكواد، بما في ذلك الأكواد الدائرية، والأكواد شبه الدائرية (QC)، والأكواد شبه الالتوائية (QT)، والأكواد شبه الدائرية العامة (GQC)، ويؤسس العلاقة بينها وبين الموديلات الفرعية لعناصر Fq[x] الحرة. يوضح البحث أن الأكواد MT هي أكواد خطية فوق PIDs، ويناقش بناء مصفوفات متعددة الحدود المولدة (GPMs)، ويستكشف خصائص مثل التماثل الذاتي والعمودية الذاتية. بالإضافة إلى ذلك، يقدم طرقًا لاشتقاق GPMs للأكواد المزدوجة ويفحص العلاقات بين القابلية للعكس، والتماثل الذاتي، والامتياز في الأكواد QC. تسهم النتائج في فهم نظرية الترميز وتطبيقاتها في نقل البيانات بشكل موثوق. [Extracted from the article]
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Databáze: Complementary Index
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Abstrakt:We begin by introducing the simple algebraic structure of cyclic, constacyclic, quasi-cyclic (QC), quasi-twisted (QT), generalized quasi-cyclic (GQC), and multi-twisted (MT) codes over finite fields. Then, we establish the correspondence between these codes and submodules of the free F<subscript>q</subscript>[x]-module (F<subscript>q</subscript>[x])<sup>l</sup>. We show that an MT code is a linear code over the principal ideal domain (PID) F<subscript>q</subscript>[x]. Hence, a basis of this code exists and is used to build a generator matrix with polynomial entries, called the generator polynomial matrix (GPM). The Hermite normal form of matrices over PIDs is exploited to achieve the reduced GPMs of MT codes. Some properties of the reduced GPM are introduced, for example, the identical equation. A formula for a GPM of the dual code of an MT code is established. At this point, special attention is paid to QC codes. We characterize GPMs for QC codes that combine reversibility and self-duality/self-orthogonality. We show the existence of binary self-orthogonal reversible QC codes that have the best known parameters as linear codes. [ABSTRACT FROM AUTHOR]
ISSN:23829761
DOI:10.22034/as.2025.19868.1629