- Marijn J.H. Heule, Manuel Kauers, and Martina Seidl, May 2019, New ways to multiply 3 x 3-matrices, Preprint
- Marijn J.H. Heule, Manuel Kauers, and Martina Seidl, July 2019, Local Search for Fast Matrix Multiplication, Proceedings of SAT'19. Preprint
- Marijn J.H. Heule, Manuel Kauers, and Martina Seidl, July 2019, A family of schemes for multiplying 3x3 matrices with 23 coefficient multiplications, Poster at ISSAC'19. Communications in Computer Algebra. Preprint

All schemes in this collection are valid for any coefficient ring. The only numbers appearing as coefficients are +1, -1, and 0. Any two schemes are inequivalent, at least when the coefficient ring is the field with two elements.

Each matrix multiplication scheme can be viewed as a \(23\times3\) table of \(3\times3\) matrices.
The *rank pattern* of a multiplication scheme is the \(23\times3\) table that contains the respective ranks.
The symmetry group of the matrix multiplication tensor permutes the rows and columns of the rank pattern,
so a suitably sorted table of the ranks can be used as an invariant of the group action.
On the left, we list our solutions separated according to this invariant.

The rank pattern names were chosen as follows.
Each column of a rank pattern is one of the 27 elements of \(\{1,2,3\}^3\).
To each of these tuples we associate a letter as indicated in the table below.
The rank pattern then corresponds to a string of 23 such letters.
As these strings tend to contain many repeated letters, we shorten them by a run length encoding.
For example, `10a2b2d2ef2jk3n` represents the string `aaaaaaaaaabbddeefjjknnn`, which in turn
encodes the following rank pattern

11111111111111111222222 |

11111111111122222111222 |

11111111112211223112222 |

Here is the assignment between letters and columns:

a | b | c | d | e | f | g | h | i | j | k | l | m | n | o | p | q | r | s | t | u | v | w | x | y | z | A |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 |

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 |

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