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Evaluation codes arising from symmetric polynomials

Barbara Gatti, Gábor Korchmáros, Gábor P. Nagy, Vincenzo Pallozzi Lavorante, Gioia Schulte

TL;DR

The paper addresses constructing and analyzing evaluation codes arising from symmetric polynomials by replacing the full symmetric polynomial space with low-dimensional linear systems. It develops a general framework via the map $\\Phi_m$ and the quotient variety $\\mathbb{F}_q^m/\\mathrm{Sym}_m$, and introduces reduced generalized Datta–Johnsen codes with lengths $N=\\frac{1}{2}q(q-1)$ and dimension $K=3$, providing lower bounds on the minimum distance that are close to optimal. For $m=2$ it derives explicit weight distributions and minimum distances for linear and quadratic polynomial cases, and proposes degree-$u$ constructions with bounds that scale with $q$, including a cubic-family construction achieving substantial distances for small $q$. In the case $m=3$, it maps distinguished points to the discriminant and determinantal varieties, deriving the reduced code $C_3'$ from linear forms on the corresponding quotient and linking code parameters to geometric section counts. Overall, the work blends finite geometry, algebraic geometry, and coding theory to produce new families of symmetric-polynomial evaluation codes with provable distance properties and ties to discriminant and determinantal structures.

Abstract

Datta and Johnsen (Des. Codes and Cryptogr., {\bf{91}} (2023), 747-761) introduced a new family of evalutation codes in an affine space of dimension $\ge 2$ over a finite field $\mathbb{F}_q$ where linear combinations of elementary symmetric polynomials are evaluated on the set of all points with pairwise distinct coordinates. In this paper, we propose a generalization by taking low dimensional linear systems of symmetric polynomials. Computation for small values of $q=7,9$ shows that carefully chosen generalized Datta-Johnsen codes $\left[\frac{1}{2}q(q-1),3,d\right]$ have minimum distance $d$ equal to the optimal value minus 1.

Evaluation codes arising from symmetric polynomials

TL;DR

The paper addresses constructing and analyzing evaluation codes arising from symmetric polynomials by replacing the full symmetric polynomial space with low-dimensional linear systems. It develops a general framework via the map and the quotient variety , and introduces reduced generalized Datta–Johnsen codes with lengths and dimension , providing lower bounds on the minimum distance that are close to optimal. For it derives explicit weight distributions and minimum distances for linear and quadratic polynomial cases, and proposes degree- constructions with bounds that scale with , including a cubic-family construction achieving substantial distances for small . In the case , it maps distinguished points to the discriminant and determinantal varieties, deriving the reduced code from linear forms on the corresponding quotient and linking code parameters to geometric section counts. Overall, the work blends finite geometry, algebraic geometry, and coding theory to produce new families of symmetric-polynomial evaluation codes with provable distance properties and ties to discriminant and determinantal structures.

Abstract

Datta and Johnsen (Des. Codes and Cryptogr., {\bf{91}} (2023), 747-761) introduced a new family of evalutation codes in an affine space of dimension over a finite field where linear combinations of elementary symmetric polynomials are evaluated on the set of all points with pairwise distinct coordinates. In this paper, we propose a generalization by taking low dimensional linear systems of symmetric polynomials. Computation for small values of shows that carefully chosen generalized Datta-Johnsen codes have minimum distance equal to the optimal value minus 1.

Paper Structure

This paper contains 15 sections, 1 theorem, 28 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Symmetric multivariable polynomials
  4. Affine varieties and curves over finite fields
  5. Upper bound on the number of points of a plane curve of Lg
  6. Mutual positions of conics in planes of odd order
  7. Linear codes
  8. Generalized Reed-Muller codes arising from multivariable polynomials over Lg
  9. The Datta-Johnsen code
  10. Evaluation codes of symmetric functions on distinguished points of the affine plane
  11. Case of linear polynomials
  12. Case of quadratic polynomials
  13. Case of polynomials of degree Lg
  14. Evaluation codes of symmetric functions on distinguished points of the affine space
  15. Acknowledgments

Key Result

Theorem 1

For odd $q\ge 7$ there exist reduced generalized Datta-Johnsen codes $\left[\frac{1}{2} q(q-1),3,D\right]$ whose minimum distance $D$ is at least $\frac{1}{2}\left(q^2-2q-2\sqrt{q}-7\right)$. The weights of the non-zero codewords fall into the interval $\left[\frac{1}{2} (q-1)-(\sqrt{q}+5),\frac{1}{

Theorems & Definitions (1)

  • Theorem 1