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arXiv · 2512.21980

A 58-Addition, Rank-23 Scheme for General 3x3 Matrix Multiplication

A. I. Perminov
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Abstract

This paper presents a new state-of-the-art algorithm for exact $3\times3$ matrix multiplication over general non-commutative rings, achieving a rank-23 scheme with only 58 scalar additions. This improves the previous best additive complexity of 60 additions without a change of basis. The result was discovered through an automated search combining ternary-restricted flip-graph exploration with greedy intersection reduction for common subexpression elimination. The resulting scheme uses only coefficients from $\{-1, 0, 1\}$, ensuring both efficiency and portability across arbitrary fields. The total scalar operation count is reduced from 83 to 81.

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A. I. Perminov. 2025-12-26. A 58-Addition, Rank-23 Scheme for General 3x3 Matrix Multiplication. https://arxiv.org/abs/2512.21980

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