Linear Codes associated to Determinantal Varieties

TL;DR

This paper studies linear codes derived from algebraic varieties defined by minors of matrices, revealing their weight distributions and properties of matrix spaces with rank constraints over finite fields.

Contribution

It introduces a class of codes from determinantal varieties, explicitly determines weight distributions for 2x2 minors, and characterizes linear spaces of matrices with rank 1.

Findings

Codes have few distinct weights

Complete weight distribution for 2x2 minors case

Maximum dimension of rank 1 matrix spaces over finite fields

Abstract

We consider a class of linear codes associated to projective algebraic varieties defined by the vanishing of minors of a fixed size of a generic matrix. It is seen that the resulting code has only a small number of distinct weights. The case of varieties defined by the vanishing of 2 x 2 minors is considered in some detail. Here we obtain the complete weight distribution. Moreover, several generalized Hamming weights are determined explicitly and it is shown that the first few of them coincide with the distinct nonzero weights. One of the tools used is to determine the maximum possible number of matrices of rank 1 in a linear space of matrices of a given dimension over a finite field. In particular, we determine the structure and the maximum possible dimension of linear spaces of matrices in which every nonzero matrix has rank 1.