It is known since the 1970s that no more than 23 multiplications are required for computing the product of two $3\times 3$-matrices. For non-commutative coefficient rings, it is not known whether it can also be done with fewer multiplications. However, there are several mutually inequivalent ways of doing the job with 23 multiplications. In this article, we extend this list considerably by providing more than 17,000 new and mutually inequivalent schemes for multiplying $3\times 3$-matrices using 23 multiplications. Moreover, we show that the set of all these schemes is a manifold of dimension at least 17.

中文翻译：

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乘以3×3矩阵的新方法

自1970年代以来就知道计算两个乘积的乘积不超过23个 $3\times 3$-矩阵。对于非交换系数环，尚不清楚是否也可以用较少的乘法来完成。但是，有23种乘法来完成这项工作的几种相互不等的方式。在本文中，我们通过提供超过17,000个新的且互不等价的乘积方案，大大扩展了此列表$3\times 3$-使用23个乘法的矩阵。此外，我们证明了所有这些方案的集合是至少17个维度的流形。