Fast matrix decomposition in F2

TL;DR

This paper introduces an efficient algorithm for block decomposition and rank computation of large dense matrices over GF(2), avoiding column permutations and improving performance compared to existing methods.

Contribution

The paper presents a novel algorithm for block decomposition of matrices over GF(2) that eliminates the need for column permutations, enhancing computational efficiency.

Findings

Algorithm outperforms existing methods in SAGE

Avoids column permutations for faster computation

Effective for large dense matrices over GF(2)

Abstract

In this work an efficient algorithm to perform a block decomposition (and so to compute the rank) of large dense rectangular matrices with entries in $\mathbb{F}_2$ is presented. Depending on the way the matrix is stored, the operations acting on rows or block of consecutive columns (stored as one integer) should be preferred. In this paper, an algorithm that completely avoids the column permutations is given. In particular, a block decomposition is presented and its running times are compared with the ones adopted into SAGE.