We present a rank-$23$ algorithm for general $3\times3$ matrix multiplication that uses $56$ additions/subtractions and $23$ multiplications, for a total of $79$ scalar operations in the standard bilinear straight-line model. This improves the recent sequence of $60$-, $59$-, and $58$-addition rank-$23$ schemes. The algorithm works over arbitrary associative, possibly noncommutative, coefficient rings. Its tensor coefficients are ternary, meaning that every coefficient lies in $\{-1,0,1\}$. Correctness is certified by the $729$ Brent equations over $\mathbb{Z}$, and the verifier also expands the straight-line program and performs additional finite-field and noncommutative implementation tests.

翻译：我们提出一个用于通用3×3矩阵乘法的秩23算法，该算法使用56次加法/减法与23次乘法，在标准双线性直线模型中总计79次标量运算。这改进了近期一系列包含60次、59次和58次加法的秩23方案。该算法适用于任意结合性（可能非交换）系数环。其张量系数为三进制，即每个系数均属于{-1,0,1}。正确性由整数环Z上的729个布伦特方程验证，验证器还对直线程序进行展开，并执行额外的有限域与非交换实现测试。