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A transform approach to polycyclic and serial codes over rings

Maryam Bajalan, Edgar Martínez-Moro, Steve Szabo

TL;DR

A transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots to study them in terms of linear algebra and invariant subspaces as well as understand the duality of the transform domain.

Abstract

In this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric.

A transform approach to polycyclic and serial codes over rings

TL;DR

A transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots to study them in terms of linear algebra and invariant subspaces as well as understand the duality of the transform domain.

Abstract

In this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant subspaces as well as understand the duality in terms of the transform domain. We also make a characterization of when two polycyclic ambient spaces are Hamming-isometric.

Paper Structure

This paper contains 10 sections, 19 theorems, 35 equations.

Table of Contents

  1. Preliminaries
  2. Finite local rings
  3. Circulant matrices
  4. The Discrete Fourier Transform over commutative rings
  5. Vandermonde matrices over commutative rings
  6. Mattson-Solomon transform and polycyclic codes over rings
  7. Polycyclic codes as invariant spaces
  8. Isometric ambient spaces
  9. Multivariable serial codes and transform domain
  10. Conclusions

Key Result

Lemma 1.1

Let $f$ be a monic polynomial in $R[x]$. Then $\mathcal{R}_f=I_1\bigoplus I_2$ where $I_1$ and $I_2$ are ideals in $\mathcal{R}_f$ if and only if there exist monic coprime polynomials $h$ and $g$ in $R[x]$ with $f=gh$ and $I_1= \langle g\rangle / \langle f\rangle$, $I_2= \langle h\rangle / \langle f

Theorems & Definitions (32)

  • Lemma 1.1: Azumaya's Lemma
  • Lemma 1.2: Theorem 3.2 in idempotents
  • Lemma 1.3: Theorem 2.1 Vel13
  • Theorem 2.1
  • Lemma 2.2: R19
  • Example 2.3
  • Example 3.1: Example 1 Cont.
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • ...and 22 more