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Some Results for Double Cyclic Codes over Fq+vFq+v2Fq.

Tenghui Deng1,2,3,4, Jing Yang1

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Entropy (Basel, Switzerland)
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This study explores double cyclic codes over finite fields with odd characteristics, focusing on their structure and generator polynomials. Researchers established relationships between these codes and their duals, leading to optimal code construction.

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double cyclic codesgenerator matricesgenerator polynomialsnon-chain rings

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Area of Science:

  • Coding Theory
  • Finite Fields
  • Algebraic Geometry

Background:

  • Double cyclic codes are a significant class of codes in coding theory, offering unique algebraic structures.
  • Understanding codes over finite non-chain rings is crucial for developing advanced error-correction techniques.
  • Previous research has explored various types of cyclic codes, but double cyclic codes over specific rings remain an active area of study.

Purpose of the Study:

  • To investigate the properties of double cyclic codes over the finite ring Fq+vFq+v2Fq, where Fq has an odd characteristic.
  • To determine the generator polynomials and generator matrices for these codes.
  • To establish the relationship between the generator polynomials of double cyclic codes and their dual codes.

Main Methods:

  • The study employs algebraic methods to analyze the structure of the ring Fq+vFq+v2Fq, noting its isomorphism to Fq×Fq×Fq.
  • Generator polynomials and matrices are derived using established coding theory techniques.
  • The generating polynomial of the dual code is computed and its relationship with the primal code's generator polynomial is established.

Main Results:

  • The paper presents a detailed analysis of double cyclic codes over the specified finite ring.
  • Explicit formulas for generator polynomials and matrices are provided.
  • A clear relationship between the generator polynomials of the double cyclic codes and their dual codes is demonstrated.

Conclusions:

  • The findings contribute to a deeper understanding of algebraic codes over finite non-chain rings.
  • The derived relationships facilitate the construction and analysis of these codes.
  • As an application, the study successfully constructs optimal codes over F3, showcasing the practical relevance of the theoretical results.