.

We now have software tools for producing canonical forms (minlex) for 16 x 16 Sudoku grids/puzzles in a very efficient manner.

These tools are derived (in my case, anyway) from the groundbreaking work of Michael Deverin (aka holdout) (see Minlex Forms with Chaining).

The essential idea is that we can determine the minlex form for any pair of rows/columns in the same band/stack, without having to check every possible transformation.

For 16x16 grids, the number of VPT's (validity-preserving transformations) applicable to a pair of rows is 24 ^ 5 = 7,962,624. There are 48 pairs (24 row pairs and 24 column pairs) that need to be considered. That's a total of 382,205,952 candidate pairs.

For 9x9 Sudoku grids, there are only 23,328 candidate pairs (1296 transformations and 18 row/col pairs).

Amazingly, however, we can get minlex forms for 16x16 grids faster than the classical canonicalisation functions (eg GridChecker) can do 9x9 grids ...

The new tools also return automorphism group size information, just like the classical 9x9 minlex functions.

My own version of "Minlex16" was developed some time back (Sept 2020), but for various reasons it is not easily distributed. But my good friend Serg has since come up with a truly "whiz-bang" (and highly original) implementation, one that is much more suitable for distribution, and I understand that he will make that available here in the near future ...

Some elementary analysis of Sudoku16 ED grid distribution is now possible, and some basic results will be listed in the next post ...

Cheers

MM