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Relative generalized Hamming weights of evaluation codes

Delio Jaramillo-Velez, Hiram H. López, Yuriko Pitones

TL;DR

The paper addresses computing relative generalized Hamming weights (RGHW) for evaluation codes by algebraically linking them to the degree of the vanishing ideal $I(X)$ and its enlargements via colon ideals. The authors prove a sharp bound: for $\mathcal{L}^2\subsetneq\mathcal{L}^1$, the $r$-th RGHW satisfies $M_r(\mathcal{L}_X^1,\mathcal{L}_X^2)=\deg(S/I)-\max_{F\in\mathcal{R}_{\prec,r}}\deg(S/(I,F))$, connecting RGHW to footprint-based degree computations and giving an operational, computable description. They also show the bound is tight in notable families, including affine Cartesian codes and squarefree evaluation codes, and they derive the next-to-minimal weights for toric codes over hypersimplices of degree $1$, including explicit weight formulas and bounds on zeros on the affine torus. The work blends Gröbner bases, affine Hilbert functions, and degree theory to extend algebraic methods to coding-theoretic invariants, with concrete computational illustrations via Macaulay2. Overall, this provides a practical, algebraic toolkit for analyzing secrecy-related parameters and weight spectra of evaluation and toric codes.

Abstract

The aim of this work is to algebraically describe the relative generalized Hamming weights of evaluation codes. We give a lower bound for these weights in terms of a footprint bound. We prove that this bound can be sharp. We compute the next-to-minimal weight of toric codes over hypersimplices of degree 1.

Relative generalized Hamming weights of evaluation codes

TL;DR

The paper addresses computing relative generalized Hamming weights (RGHW) for evaluation codes by algebraically linking them to the degree of the vanishing ideal and its enlargements via colon ideals. The authors prove a sharp bound: for , the -th RGHW satisfies , connecting RGHW to footprint-based degree computations and giving an operational, computable description. They also show the bound is tight in notable families, including affine Cartesian codes and squarefree evaluation codes, and they derive the next-to-minimal weights for toric codes over hypersimplices of degree , including explicit weight formulas and bounds on zeros on the affine torus. The work blends Gröbner bases, affine Hilbert functions, and degree theory to extend algebraic methods to coding-theoretic invariants, with concrete computational illustrations via Macaulay2. Overall, this provides a practical, algebraic toolkit for analyzing secrecy-related parameters and weight spectra of evaluation and toric codes.

Abstract

The aim of this work is to algebraically describe the relative generalized Hamming weights of evaluation codes. We give a lower bound for these weights in terms of a footprint bound. We prove that this bound can be sharp. We compute the next-to-minimal weight of toric codes over hypersimplices of degree 1.
Paper Structure (5 sections, 17 theorems, 58 equations, 1 figure)

This paper contains 5 sections, 17 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An algebraic representation
  4. Toric codes over hypersimplices
  5. Examples

Key Result

Lemma 2.1

cocoa-book Let $X$ be a finite subset of $\mathbb{A}^s$, let $P$ be a point in $X$, $P=(p_1,\ldots,p_s)$, and let $I_{P}$ be the vanishing ideal of $P$. Then $I_P$ is a maximal ideal of height $s$, and $I(X)=\bigcap_{P\in X}I_{P}$ is the primary decomposition of $I(X)$.

Figures (1)

  • Figure 1: Lattice points defining $\mathcal{L}^{1}$ and $\mathcal{L}^{2}$ of Example \ref{['sharp-bound-inequ']}.

Theorems & Definitions (32)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Definition 2.5
  • Lemma 2.6
  • Remark 2.7
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 22 more