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On the number of minimal and next-to-minimal weight codewords of toric codes over hypersimplices

Cícero Carvalho, Nupur Patanker

TL;DR

This paper advances the understanding of toric codes $\mathcal{C}(d)$ associated with hypersimplices by precisely characterizing and counting the minimal and next-to-minimal weight codewords for the regime $3 \le d < s$. It leverages monomial equivalence between $\mathcal{C}(d)$ and $\mathcal{C}(s-d)$ in key ranges and employs Gröbner-basis footprint techniques to identify exact polynomial forms that achieve minimum and next-to-minimal weights, along with explicit weight- and count-formulas. The main contributions include closed-form counts for minimal-weight codewords in the $2d \le s$ and $s<2d<2s$ regimes and a detailed description and enumeration of next-to-minimal-weight codewords for $2d+2 \le s$ (with duality-based extensions for $2d-2 \ge s$). These results enhance weight enumerator knowledge, provide geometric interpretations via hypersurface configurations in the affine torus, and have implications for error-detection performance of these toric codes.

Abstract

Toric codes are a type of evaluation code introduced by J.P. Hansen in 2000. They are produced by evaluating (a vector space composed by) polynomials at the points of $(\mathbb{F}_q^*)^s$, the monomials of these polynomials being related to a certain polytope. Toric codes related to hypersimplices are the result of the evaluation of a vector space of homogeneous monomially square-free polynomials of degree $d$. The dimension and minimum distance of toric codes related to hypersimplices have been determined by Jaramillo et al. in 2021. The next-to-minimal weight in the case $d = 1$ has been determined by Jaramillo-Velez et al. in 2023, and has been determined in the cases where $3 \leq d \leq \frac{s - 2}{2}$ or $\frac{s + 2}{2} \leq d < s$, by Carvalho and Patanker in 2024. In this work we characterize and determine the number of minimal (respectively, next-to-minimal) weight codewords when $3 \leq d < s$ (respectively, $3 \leq d \leq \frac{s - 2}{2}$ or $\frac{s + 2}{2} \leq d < s$).

On the number of minimal and next-to-minimal weight codewords of toric codes over hypersimplices

TL;DR

This paper advances the understanding of toric codes associated with hypersimplices by precisely characterizing and counting the minimal and next-to-minimal weight codewords for the regime . It leverages monomial equivalence between and in key ranges and employs Gröbner-basis footprint techniques to identify exact polynomial forms that achieve minimum and next-to-minimal weights, along with explicit weight- and count-formulas. The main contributions include closed-form counts for minimal-weight codewords in the and regimes and a detailed description and enumeration of next-to-minimal-weight codewords for (with duality-based extensions for ). These results enhance weight enumerator knowledge, provide geometric interpretations via hypersurface configurations in the affine torus, and have implications for error-detection performance of these toric codes.

Abstract

Toric codes are a type of evaluation code introduced by J.P. Hansen in 2000. They are produced by evaluating (a vector space composed by) polynomials at the points of , the monomials of these polynomials being related to a certain polytope. Toric codes related to hypersimplices are the result of the evaluation of a vector space of homogeneous monomially square-free polynomials of degree . The dimension and minimum distance of toric codes related to hypersimplices have been determined by Jaramillo et al. in 2021. The next-to-minimal weight in the case has been determined by Jaramillo-Velez et al. in 2023, and has been determined in the cases where or , by Carvalho and Patanker in 2024. In this work we characterize and determine the number of minimal (respectively, next-to-minimal) weight codewords when (respectively, or ).

Paper Structure

This paper contains 3 sections, 8 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Characterization and number of minimum weight codewords
  3. Characterization and number of next-to-minimal weight codewords

Key Result

Proposition 2.1

car-nupur Let $f \in \mathbb{F}_q[\boldsymbol{t}]$ be a homogeneous, monic, monomially square-free polynomial of degree $d$, such that $\textrm{LM}(f) = t_1 . \cdots. t_d$, and assume that $2d \leq s$. Then $\varphi(f + I_X)$ is a minimum weight codeword if and only if $f = (t_1 + \alpha_{1} t_{c_1}

Theorems & Definitions (10)

  • Definition 1.1
  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Example 3.5