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On the Algebraic Structure Underlying the Support Enumerators of Linear Codes

Nitin Kenjale, Anuradha S. Garge

TL;DR

The work introduces coordinate-level invariants—the support distribution $S_i$ and the support enumerator $S_C(z)$—to refine the classical weight-based view of linear codes. It develops a MacWilliams-type identity linking the normalized support enumerators of a code $C$ and its dual $C^ot$, revealing how coordinate activity transforms under duality and highlighting coordinates with special roles (e_i). A direct consequence is a necessary self-duality criterion expressed purely in terms of equality of support distributions. The authors illustrate the framework with concrete codes (simplex, Hamming, repetition, and extended Hamming), demonstrating both the algebraic structure and potential for applications in local error analysis and symmetry studies.

Abstract

In this paper, we have introduced the concepts of support distribution and the support enumerator as refinements of the classical weight distribution and weight enumerator respectively, capturing coordinate level activity in linear block codes. More precisely, we have established formula for counting codewords in the linear code C whose i-th coordinate is nonzero. Moreover, we derived a MacWilliam's type identity, relating the normalized support enumerators of a linear code and its dual, explaining how coordinate information transforms under duality. Using this identity we deduce a condition for self duality based on the equality of support distributions. These results provide a more detailed understanding of code structure and complement classical weight based duality theory.

On the Algebraic Structure Underlying the Support Enumerators of Linear Codes

TL;DR

The work introduces coordinate-level invariants—the support distribution and the support enumerator —to refine the classical weight-based view of linear codes. It develops a MacWilliams-type identity linking the normalized support enumerators of a code and its dual , revealing how coordinate activity transforms under duality and highlighting coordinates with special roles (e_i). A direct consequence is a necessary self-duality criterion expressed purely in terms of equality of support distributions. The authors illustrate the framework with concrete codes (simplex, Hamming, repetition, and extended Hamming), demonstrating both the algebraic structure and potential for applications in local error analysis and symmetry studies.

Abstract

In this paper, we have introduced the concepts of support distribution and the support enumerator as refinements of the classical weight distribution and weight enumerator respectively, capturing coordinate level activity in linear block codes. More precisely, we have established formula for counting codewords in the linear code C whose i-th coordinate is nonzero. Moreover, we derived a MacWilliam's type identity, relating the normalized support enumerators of a linear code and its dual, explaining how coordinate information transforms under duality. Using this identity we deduce a condition for self duality based on the equality of support distributions. These results provide a more detailed understanding of code structure and complement classical weight based duality theory.
Paper Structure (8 sections, 6 theorems, 36 equations)

This paper contains 8 sections, 6 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Support Framework
  4. Relating Support Enumerators of $C$ and $C^\perp$
  5. Necessary condition for Self-Duality
  6. Illustrations of Theorems
  7. Illustration of support enumerator identity
  8. Illustration of Necessary condition for self-duality

Key Result

Lemma 1.1

McE Let $C\subseteq\mathbb{F}_q^n$ be a linear code and $C^\perp$ be its dual code. Let $\chi:\mathbb{F}_q\to\mathbb{C}^*$ be a nontrivial additive character. For each $u\in\mathbb{F}_q^n,$ define $S(u)=\sum_{c\in C}\chi(u\cdot c^t).$ Then

Theorems & Definitions (14)

  • Lemma 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • ...and 4 more