A robust algorithm for $k$-point grid generation and symmetry reduction

We develop an algorithm for i) computing generalized regular $k$-point grids, ii) reducing the grids to their symmetrically distinct points, and iii) mapping the reduced grid points into the Brillouin zone. The algorithm exploits the connection between integer matrices and finite groups to achieve a computational complexity that is linear with the number of $k$-points. The favorable scaling means that, at a given $k$-point density, all possible commensurate grids can be generated (as suggested by Moreno and Soler) and quickly reduced to identify the grid with the fewest symmetrically unique $k$-points. These optimal grids provide significant speed-up compared to Monkhorst-Pack $k$-point grids; they have better symmetry reduction resulting in fewer irreducible $k$-points at a given grid density. The integer nature of this new reduction algorithm also simplifies issues with finite precision in current implementations. The algorithm is available as open source software.