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Structure of weighted projective Reed-Muller codes

Jade Nardi, Rodrigo San-José

Abstract

We provide a comprehensive overview of the fundamental structural properties of weighted projective Reed-Muller codes. We give a recursive construction for these codes, under some conditions for the weights, and we use it to derive bounds on the generalized Hamming weights and to obtain a recursive construction for their subfield subcodes and their dual codes. The dual codes are further studied in more generality, where the recursive constructions may not apply, obtaining a description as an evaluation code when the degree is low. We also provide insights into the Schur products of these codes when they are not degenerate.

Structure of weighted projective Reed-Muller codes

Abstract

We provide a comprehensive overview of the fundamental structural properties of weighted projective Reed-Muller codes. We give a recursive construction for these codes, under some conditions for the weights, and we use it to derive bounds on the generalized Hamming weights and to obtain a recursive construction for their subfield subcodes and their dual codes. The dual codes are further studied in more generality, where the recursive constructions may not apply, obtaining a description as an evaluation code when the degree is low. We also provide insights into the Schur products of these codes when they are not degenerate.

Paper Structure

This paper contains 11 sections, 34 theorems, 106 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear codes
  4. Weighted projective spaces
  5. Weighted projective Reed-Muller codes
  6. Weighted projective Reed-Solomon codes
  7. Recursive construction of WPRM codes
  8. Examples
  9. Duals
  10. Representation as monomial codes
  11. Schur products of WPRM codes

Key Result

Lemma 2.5

Let $w= (w_0,\dots,w_m)$ and let $\gamma=\gcd(w_0,\dots,w_m)$. Set $w/\gamma=(w_0/\gamma,\dots,w_m/\gamma)$. Then we have ${\mathbb{P}}(w)(\mathbb{F}_q)={\mathbb{P}}(w/\gamma)(\mathbb{F}_q)$ and

Figures (1)

  • Figure 1: Tree of possible weight reductions (via Corollaries \ref{['c:codes_weightswithgcd']} and \ref{['c:codes_delorme']}), and decompositions (via Theorem \ref{['t:rec1weakcond']}) corresponding to Example \ref{['ex:tree']}.

Theorems & Definitions (86)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7: (Delorme's weight reduction)
  • Lemma 2.8
  • proof
  • Remark 2.9
  • ...and 76 more