Fast Matrix Multiplication in Small Formats: Discovering New Schemes with an Open-Source Flip Graph Framework

Abstract

An open-source C++ framework for discovering fast matrix multiplication schemes using the flip graph approach is presented. The framework supports multiple coefficient rings -- binary ($\mathbb{Z}_2$), modular ternary ($\mathbb{Z}_3$) and integer ternary ($\mathbb{Z}_T = \{-1,0,1\}$) -- and implements both fixed-dimension and meta-dimensional search operators. Using efficient bit-level encoding of coefficient vectors and OpenMP parallelism, the tools enable large-scale exploration on commodity hardware. The study covers 680 schemes ranging from $(2 \times 2 \times 2)$ to $(16 \times 16 \times 16)$, with 276 schemes now in $\mathbb{Z}_T$ coefficients and 117 in integer coefficients. With this framework, the multiplicative complexity (rank) is improved for 79 matrix multiplication schemes. Notably, a new $4 \times 4 \times 10$ scheme requiring only 115 multiplications is discovered, achieving $\omega \approx 2.80478$ and beating Strassen's exponent for this specific size. Additionally, 93 schemes are rediscovered in ternary coefficients that were previously known only over rationals or integers, and 68 schemes in integer coefficients that previously required fractions. All tools and discovered schemes are made publicly available to enable reproducible research.