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On the structure of repeated-root polycyclic codes over local rings

Maryam Bajalan, Edgar Martinez-Moro, Reza Sobhani, Steve Szabo, Gulsum Gozde Yilmazguc

Abstract

This paper provides the Generalized Mattson Solomon polynomial for repeated-root polycyclic codes over local rings that gives an explicit decomposition of them in terms of idempotents that completes the single root study. It also states some structural properties of repeated-root polycyclic codes over finite fields in terms of matrix product codes. Both approaches provide a description of the $\perp_0$-dual code of a given polycyclic code.

On the structure of repeated-root polycyclic codes over local rings

Abstract

This paper provides the Generalized Mattson Solomon polynomial for repeated-root polycyclic codes over local rings that gives an explicit decomposition of them in terms of idempotents that completes the single root study. It also states some structural properties of repeated-root polycyclic codes over finite fields in terms of matrix product codes. Both approaches provide a description of the -dual code of a given polycyclic code.
Paper Structure (14 sections, 30 theorems, 56 equations)

This paper contains 14 sections, 30 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hasse derivative and Generalized Discrete Fourier Transform
  4. Generalized Vandermonde matrices
  5. Generalized Mattson Solomon polynomial
  6. The decomposition of the ambient space
  7. Decomposition of the codes
  8. $\perp_0$ duality
  9. A note on multivariable codes
  10. Matrix-Product Structure of Certain Polycyclic Codes
  11. Duality
  12. Matrix-product codes
  13. The isomorphism
  14. Proof

Key Result

Lemma 2.1

Let $p(x)$ and $q(x)$ be two polynomials in $R[x].$

Theorems & Definitions (59)

  • Lemma 2.1
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 3.1
  • proof
  • Example 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 49 more