Hyperplane Sections of Determinantal Varieties over Finite Fields and Linear Codes

TL;DR

This paper investigates the number of rational points on hyperplane sections of determinantal varieties over finite fields, explores sections with maximum points, and relates findings to properties of associated linear codes.

Contribution

It generalizes and expands previous results on rational points and linear codes related to determinantal varieties, including weight distribution and minimum distance.

Findings

Identified hyperplane sections with maximum rational points.

Determined the number of rational points on various sections.

Connected geometric properties to linear code parameters.

Abstract

We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify sections giving the maximum number of ${\mathbb{F}}_q$-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance and generalized Hamming weights. The previously known results about these are generalized and expanded significantly