Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Enumeration of minimum weight codewords of affine Cartesian codes

Sakshi Dang, Sudhir R. Ghorpade

TL;DR

The article develops explicit formulas for counting minimum-weight codewords of affine Cartesian codes in the nested-subfield setting. By modeling the action of the affine group Aff$( ext{A})$ via a block lower-triangular structure and applying the orbit-stabilizer principle, it enumerates minimum-weight codewords across all admissible orders $u$ and unifies the weight-distribution results for Reed-Solomon and Reed-Muller codes as special cases. The main contribution is a complete, case-by-case enumeration in terms of the field sizes, subfield multiplicities, and $q$-binomial coefficients, enriching the structural understanding of affine Cartesian codes and their automorphism groups. The methods and formulas provide a unified framework for weight enumeration in a broad class of evaluation codes arising from Cartesian products of nested subfields, with potential implications for decoding and code-design applications.

Abstract

Affine Cartesian codes were first discussed by Geil and Thomsen in 2013 in a broader framework and were formally introduced by López, Rentería-Márquez and Villarreal in 2014. These are linear error-correcting codes obtained by evaluating polynomials at points of a Cartesian product of subsets of the given finite field. They can be viewed as a vast generalization of Reed-Muller codes. In 1970, Delsarte, Goethals and MacWilliams gave a %characterization of minimum weight codewords of Reed-Muller codes and also formula for the minimum weight codewords of Reed-Muller codes. Carvalho and Neumann in 2020 considered affine Cartesian codes in a special setting where the subsets in the Cartesian product are nested subfields of the given finite field, and gave a characterization of their minimum weight codewords. We use this to give an explicit formula for the number of minimum weight codewords of affine Cartesian codes in the case of nested subfields. This is seen to unify the known formulas for the number of minimum weight codewords of Reed-Solomon codes and Reed-Muller codes.

Enumeration of minimum weight codewords of affine Cartesian codes

TL;DR

The article develops explicit formulas for counting minimum-weight codewords of affine Cartesian codes in the nested-subfield setting. By modeling the action of the affine group Aff via a block lower-triangular structure and applying the orbit-stabilizer principle, it enumerates minimum-weight codewords across all admissible orders and unifies the weight-distribution results for Reed-Solomon and Reed-Muller codes as special cases. The main contribution is a complete, case-by-case enumeration in terms of the field sizes, subfield multiplicities, and -binomial coefficients, enriching the structural understanding of affine Cartesian codes and their automorphism groups. The methods and formulas provide a unified framework for weight enumeration in a broad class of evaluation codes arising from Cartesian products of nested subfields, with potential implications for decoding and code-design applications.

Abstract

Affine Cartesian codes were first discussed by Geil and Thomsen in 2013 in a broader framework and were formally introduced by López, Rentería-Márquez and Villarreal in 2014. These are linear error-correcting codes obtained by evaluating polynomials at points of a Cartesian product of subsets of the given finite field. They can be viewed as a vast generalization of Reed-Muller codes. In 1970, Delsarte, Goethals and MacWilliams gave a %characterization of minimum weight codewords of Reed-Muller codes and also formula for the minimum weight codewords of Reed-Muller codes. Carvalho and Neumann in 2020 considered affine Cartesian codes in a special setting where the subsets in the Cartesian product are nested subfields of the given finite field, and gave a characterization of their minimum weight codewords. We use this to give an explicit formula for the number of minimum weight codewords of affine Cartesian codes in the case of nested subfields. This is seen to unify the known formulas for the number of minimum weight codewords of Reed-Solomon codes and Reed-Muller codes.
Paper Structure (14 sections, 20 theorems, 243 equations)

This paper contains 14 sections, 20 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Affine Cartesian codes
  4. Affine Cartesian codes for Nested Subfields
  5. A Characterization of the Affine Group
  6. Enumeration of minimum weight codewords
  7. Case $\ell = d_{j+1}-1$
  8. Case $\ell < d_{j+1}-1$
  9. Subcase $t_k =r$
  10. Subcase $t_k <r$
  11. Proof of the Main Theorem
  12. Reed-Solomon codes
  13. Reed-Muller codes
  14. Examples

Key Result

Theorem 1.1

Let $m \geq 1$ and ${\mathcal{A}} = F_1 \times \cdots \times F_1 \times \cdots \times F_{\lambda} \times \cdots \times F_{{\lambda}} = F_1^{\mu_1} \times \cdots \times F_{{\lambda}}^{\mu_{{\lambda}}} \subseteq \hbox{$\mathbb F$}_q^m$ as before. Then For $1 \leq u \leq K= \sum\limits_{i=1}^{m} (d_i-1)$, write where $j, \ell$ are uniquely determined integers such that $0 \leq j < m$ and $0 < \ell

Theorems & Definitions (47)

  • Theorem 1.1
  • Lemma 2.1: Hou, Lemma 2.21
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Dummit_Foote, Chapter 4, Proposition 2
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 37 more