[wbhart]

As already noted in a post by fdej further down, Waksman's algorithm from 1970, which works over the complex numbers, requires only 46 multiplications (and I guess, divisions by 2, which may or may not be relevant depending on your actual ring).

[nemonemo]

The answer says "For rings in which division by 2 is permitted". Is there the same constraint for AlphaEvolve's algorithm?

Edit2: Z_2 has characteristics 2.

Edit: AlphaEvolve claims it works over any field with characteristic 0. It appears Waksman's could be an existing work. From the AlphaEvolve paper: "For 56 years, designing an algorithm with fewer than 49 multiplications over any field with characteristic 0 was an open problem. AlphaEvolve is the first method to find an algorithm to multiply two 4 × 4 complex-valued matrices using 48 multiplications."

[fdej]

If you don't want to allow division by 2 then there is Winograd's algorithm from 1967 which works over any commutative ring and uses 48 multiplications for 4 x 4.

[wbhart]

Z_2 has characteristic 2, not 0.