Revisiting “Rapid multiplication of rectangular matrices”

What is the largest constant α such that for all ɛ > 0, an n × n and (Formula presented.) matrices can be multiplied in (Formula presented.) ? Coppersmith (1982) was the first to present a lower bound on α, showing that (Formula presented.). Coppersmith further proposes a second proof that leads to the bound (Formula presented.), and combined with a newer construction by Coppersmith and Winograd (1982), the second bound improves to (Formula presented.). We revisit this work, show that the second proof is incorrect, and propose an alternative one. Our alternative proof uses a recursive construction, based on the Schönhage’s multiplication stacking technique, that converges to the same bound on α.