In this work we describe an efficient implementation of a hierarchy of
algorithms for Gaussian elimination upon dense matrices over the field with two
elements. We discuss both well-known and new algorithms as well as our
implementations in the M4RI library, which has been adopted into Sage. The
focus of our discussion is a block iterative algorithm for PLE decomposition
which is inspired by the M4RI algorithm. The implementation presented in this
work provides considerable performance gains in practice when compared to the
previously fastest implementation. We provide performance figures on x86_64
CPUs to demonstrate the alacrity of our approach