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

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.