Generalized Hamming weights of toric codes over hypersimplices and square-free affine evaluation codes

Abstract

Let ${\mathbb{F}}_q$ be a finite field with $q$ elements, where $q$ is a power of prime $p$. A polynomial over ${\mathbb{F}}_q$ is square-free if all its monomials are square-free. In this note, we determine an upper bound on the number of zeroes in the affine torus $T=({\mathbb{F}}_q^{\ast }{)}^s$ of any set of $r$ linearly independent square-free polynomials over ${\mathbb{F}}_q$ in $s$ variables, under certain conditions on $r$, $s$ and degree of these polynomials. Applying the results, we partly obtain the generalized Hamming weights of toric codes over hypersimplices and square-free evaluation codes, as defined in \cite{hyper}. Finally, we obtain the dual of these toric codes with respect to the Euclidean scalar product.

Due to an error in the proof, Lemma 4.6 to Lemma 4.9 has been deleted