Exponential sums and rank of persymmetric matrices over F_2

TL;DR

This paper develops a method to explicitly count the number of persymmetric matrices over F_2 with a given rank, enhancing understanding of their structure and properties.

Contribution

It introduces a novel approach for deriving explicit formulas for the rank distribution of persymmetric matrices over F_2.

Findings

Explicit formulas for counting rank i persymmetric matrices over F_2

A new method for analyzing matrix rank distributions over finite fields

Enhanced understanding of the structure of persymmetric matrices

Abstract

Over the finite field with two elements, we present a method for obtaining explicit expressions for the number of rank i matrices of the form A above B, where A is persymmetric (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j = r+s).