A fast search algorithm for 〈m,m,m〉 Triple Product Property triples and an application for 5×5 matrix multiplication

Abstract We present a new fast search algorithm for 〈m,m,m〉 Triple Product Property (TPP) triples as defined by Cohn and Umans in 2003. The new algorithm achieves a speed-up factor of 40 up to 194 in comparison to the best known search algorithm. With a parallelized version of the new algorithm we are able to search for TPP triples in groups up to order 55. As an application we identify lists “C1” and “C2” of groups that, if they contain a 〈5,5,5〉 TPP triple, could realize 5×5 matrix multiplication with under 100, respectively under 125, scalar multiplications, i.e., the best known upper bound by Makarov (1987), respectively the trivial upper bound. With our new algorithm we show that no group in this list can realize 5×5 matrix multiplication better than Makarov's algorithm. We also show a direction towards a modified group-theoretic search, not covered by the C1 list.