Generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes

Nupur Patanker; Sanjay Kumar Singh

doi:10.3934/amc.2021013

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2023, Volume 17, Issue 3: 626-643. Doi: 10.3934/amc.2021013

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Generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes

Nupur Patanker^, and
Sanjay Kumar Singh

Indian Institute of Science Education and Research, Bhopal, India

^* Corresponding author: Nupur Patanker
^* Corresponding author: Nupur Patanker

Received: December 2020

Revised: March 2021

Early access: May 2021

Published: June 2023

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Abstract

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements, where $q$ is a power of a prime $p$ . A polynomial over $\mathbb{F}_{q}$ is monomially squarefree if all its monomials are squarefree. In this paper, we determine an upper bound on the number of common zeroes of any set of $r$ linearly independent monomially squarefree polynomials of $\mathbb{F}_{q}[t_{1}, t_{2}, \dots, t_{s}]$ in the affine torus $T = (\mathbb{F}_{q}^{*})^{s}$ under certain conditions on $r$ , $s$ and the degree of these polynomials. Applying the results, we obtain the generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes, as defined in [14].

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Mathematics Subject Classification: Primary: 13P25; Secondary: 14G50, 94B27, 11T71, 06A07.

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