Description of fast matrix multiplication algorithm: ⟨4×4×4:48⟩

Algorithm type

16 X^{2} Y^{2} Z^{2} + 32 X Y Z

Algorithm definition

Algorithm description

Algorithm symmetries

The following group of 64 isotropies acts as a permutation group on algorithm tensor representation:

maple tensor representation (compressed by bzip2);

These encodings are given in compressed text format using the maple computer algebra system. In each cases, the last line could be understood as a description of the encoding with respect to classical matrix multiplication algorithm. As these outputs are structured, one can construct easily a parser to its favorite format using the maple documentation without this software.

Commentaries

The original algorithm introduced by Novikov et ali (AlphaEvolve 2025) involves complex-valued multiplications.

The variant of this algorithm presented here matches the current best-known tensor rank while removing the reliance on complex coefficients. It is taken from (Dumas, Pernet and Sedoglavic 2025).

A variant with the currently best known complexity $(7 + \frac{3}{8}) n^{2 + {log}_{4} 3} + o (n^{2 + {log}_{4} 3})$ is presented in (Dumas, Pernet, Sedoglavic, Stapleton and Tichavský 2026).

Another more accurate variant is presented in (Dumas, Pernet and Sedoglavic 2026) and its currently best known complexity is $(9 + \frac{3}{16}) n^{2 + {log}_{4} 3} + o (n^{2 + {log}_{4} 3})$ .