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

2026.

Perminov Andrew Igorevich

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 (Z2), modular ternary (Z3) and integer ternary (ZT={−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 (2×2×2) to (16×16×16), with 276 schemes now in ZT coefficients and 117 in integer coefficients. With this framework, the multiplicative complexity (rank) is improved for 79 matrix multiplication schemes. Notably, a new 4×4×10 scheme requiring only 115 multiplications is discovered, achieving ω≈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.

Keywords: тензорный ранг, tensor rank, быстрое матричное умножение, флип граф, fast matrix multiplication, flip graph, ternary integer coefficient set, тернарное множество коэффициентов