Evaluation codes arising from symmetric polynomials

TL;DR

This paper generalizes a class of evaluation codes based on symmetric polynomials, showing through computations that certain codes nearly achieve optimal minimum distance.

Contribution

It introduces a generalized construction of evaluation codes from symmetric polynomials and demonstrates their near-optimal minimum distance through computational evidence.

Findings

Codes with parameters [1/2 q(q-1), 3, d] have minimum distance close to optimal.

Generalized codes outperform previous constructions in certain parameters.

Computational results for q=7,9 support the effectiveness of the generalization.

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.