The second generalized Hamming weight of some evaluation codes arising from a projective torus

TL;DR

This paper determines the second generalized Hamming weight of evaluation codes from a projective torus, enabling the computation of these weights for codes associated with complete bipartite graphs, and explores related code properties.

Contribution

It introduces methods to compute the second generalized Hamming weight for codes from a projective torus and extends results to codes from complete intersections with known minimum distances.

Findings

Second generalized Hamming weight computed for codes from a projective torus

Results applied to codes parameterized by complete bipartite graph edges

Example provided for complete weight hierarchy determination

Abstract

In this paper we find the second generalized Hamming weight of some evaluation codes arising from a projective torus, and it allows to compute the second generalized Hamming weight of the codes parameterized by the edges of any complete bipartite graph. Also, at the beginning, we obtain some results about the generalized Hamming weights of some evaluation codes arising from a complete intersection when the minimum distance is known and they are non--degenerate codes. Finally we give an example where we use these results to determine the complete weight hierarchy of some codes.