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Using a naive lower bound and schoolbook matrix multiplication for the upper bound, one can straightforwardly conclude that {{math|2 ≤ ω ≤ 3}}. Whether {{math|1=ω = 2}} is a major open question in [[theoretical computer science]], and there is a line of research developing matrix multiplication algorithms to get improved bounds on {{math|ω}}.
All recent algorithms in this line of research use the ''laser method'', a generalization of the Coppersmith–Winograd algorithm, which was given by [[Don Coppersmith]] and [[Shmuel Winograd]] in 1990 and was the best matrix multiplication algorithm until 2010.<ref name="coppersmith">{{cite journal|doi=10.1016/S0747-7171(08)80013-2 |title=Matrix multiplication via arithmetic progressions |url=http://www.cs.umd.edu/~gasarch/TOPICS/ramsey/matrixmult.pdf |year=1990 |last1=Coppersmith |first1=Don |last2=Winograd |first2=Shmuel |journal=Journal of Symbolic Computation |volume=9|issue=3|pages=251|doi-access=free }}</ref> The conceptual idea of these algorithms is similar to Strassen's algorithm: a method is devised for multiplying two {{math|''k'' × ''k''}}-matrices with fewer than {{math|''k''<sup>3</sup>}} multiplications, and this technique is applied recursively. The laser method has limitations to its power: [[Andris Ambainis|Ambainis]], Filmus and [[Jean-François Le Gall|Le Gall]] prove that it cannot be used to show that {{math|ω < 2.3725}} by analyzing higher and higher tensor powers of a certain identity of Coppersmith and Winograd and neither {{math|ω < 2.3078}} for a wide class of variants of this approach.<ref name="afl142">{{Cite book |last1=Ambainis |first1=Andris |title=Proceedings of the forty-seventh annual ACM symposium on Theory of Computing |last2=Filmus |first2=Yuval |last3=Le Gall |first3=François |date=2015-06-14 |publisher=Association for Computing Machinery |isbn=978-1-4503-3536-2 |series=STOC '15 |___location=Portland, Oregon, USA |pages=585–593 |chapter=Fast Matrix Multiplication |doi=10.1145/2746539.2746554 |chapter-url=https://doi.org/10.1145/2746539.2746554 |arxiv=1411.5414 |s2cid=8332797}}</ref> In 2022 Duan, Wu and Zhou devised a variant breaking the first of the two barriers with {{math|ω < 2.37188}},<ref name="dwz22" /> they do so by identifying a source of potential optimization in the laser method termed ''combination loss'' for which they compensate using an asymmetric version of the hashing method in the Coppersmith–Winograd algorithm. Nonetheless, the above are a classical example of [[Galactic algorithm#Matrix multiplication|galactic algorithms]]. On the opposite, Victor Pan proposed feasible matrix multiplication algorithms with an exponent slightly above 2.77, but in return with a much smaller hidden constant coefficient. <ref>{{Citation | last1=Laderman | first1=Julian | last2=Pan | first2=Victor |last3=Sha | first3=Xuan-He | title=On practical algorithms for accelerated matrix multiplication | year=1992 | journal=Linear Algebra and its Applications | volume=162-164 | pages=557-588 | doi=10.1016/0024-3795(92)90393-O}}</ref>, <ref>{{Citation | last1=Respondek | first1=Jerzy S. | title=Correction of ‘J. Laderman, V. Pan, X.–H. Sha, On practical Algorithms for Accelerated Matrix Multiplication, Linear Algebra and its Applications. Vol. 162-164 (1992) pp. 557-588’ | year=2024 | journal=Linear and Multilinear Algebra | pages=1-11 | doi=10.1080/03081087.2024.2391807 }}</ref>
=== Group theory reformulation of matrix multiplication algorithms ===
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