Evaluation codes arising from symmetric polynomials

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

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arXiv · 2502.14414

Evaluation codes arising from symmetric polynomials

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

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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.

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Barbara Gatti, Gábor Korchmáros, Gábor P. Nagy, Vincenzo Pallozzi Lavorante, Gioia Schulte. 2025-02-20. Evaluation codes arising from symmetric polynomials. https://arxiv.org/abs/2502.14414

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