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 $ω\approx 2.80478$ 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.

翻译：本文提出一个基于翻转图方法发现快速矩阵乘法方案的开源C++框架。该框架支持多种系数环——二元（$\mathbb{Z}_2$）、模三元（$\mathbb{Z}_3$）和整三元（$\mathbb{Z}_T = \{-1,0,1\}$）——并实现了固定维度和元维度两种搜索算子。通过高效的系数向量位级编码和OpenMP并行技术，该工具可在商用硬件上实现大规模探索。研究覆盖了从$(2\times2\times2)$到$(16\times16\times16)$的680种方案，其中276种采用$\mathbb{Z}_T$系数，117种采用整数系数。利用该框架，79个矩阵乘法方案的乘法复杂度（秩）得到改进。值得注意的是，发现了一个仅需115次乘法的新$4\times4\times10$方案，其指数$ω\approx 2.80478$，在此特定规模下超越了Strassen指数。此外，有93种之前仅在有理数或整数系数下已知的方案在三元系数中被重新发现，68种之前需要分数系数的方案在整数系数中被发现。所有工具和发现的方案均已公开，以支持可重复研究。