Four Inequivalent Ternary Rank-90 Decompositions of ⟨3,5,8⟩ over ℤ

Abstract

We report four mutually inequivalent exact algorithms for the product of a 3×5 by a 5×8 matrix, each using 90 multiplications in which every coefficient is −1, 0 or +1. Rank 90 is the best rank published for ⟨3,5,8⟩ in any coefficient domain; the rank-90 scheme in the public catalogue carries coefficients up to absolute value 4. To the best of our knowledge these are the first ternary decompositions at the record rank for this format over ℤ. Their de Groote profiles are pairwise distinct and none of them matches the catalogued solution, so the four schemes are inequivalent to one another and to the published entry under the sandwiching action of GL3(ℤ)×GL5(ℤ)×GL8(ℤ): four new orbits, not rescalings of an existing solution. Each scheme reconstructs all 14 400 entries of the matrix-multiplication tensor with zero residual over the integers and multiplies integer matrices correctly under direct evaluation. The coefficient archives are committed by SHA-256 in the paper, and a live verification endpoint re-derives every claim inside an isolated container.