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Let $A$ and $B$ be $n \times n$ matrices. Strassen's algorithm for multiplication works on a partitioning of $A$ and $B$ into $2^2$ submatrices each. Is there any generalization of Strassen's algorithm where the partitioning is done into, say, $k^2$ submatrices?

Note: What I mean by a $3^2$ partitioning of $A$ is: $$A = \left(\begin{array}{ccc} A_{1,1} & A_{1,2} & A_{1,3} \\ A_{2,1} & A_{2,2} & A_{2,3} \\ A_{3,1} & A_{3,2} & A_{3,3} \end{array} \right) $$ where each of $A_{i,j}$ is an $\frac{n}{3} \times \frac{n}{3}$ matrix.

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Yes, in some cases. See for instance https://web.archive.org/web/20120912032909/https://www.csd.uwo.ca/~mislam63/ms_thesis.pdf

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