Some concerns on the border rank of Kronecker products of the Coppersmith-Winograd tensor

TL;DR

This paper proves the exact border rank of the Kronecker square of the Coppersmith-Winograd tensor and discusses potential extensions to higher Kronecker powers, contributing to tensor rank complexity understanding.

Contribution

Provides a detailed proof of the border rank of the Kronecker square of the Coppersmith-Winograd tensor and explores extensions to higher powers.

Findings

Border rank of the Kronecker square is (q+2)^2

Indicates potential border rank of (q+2)^m for m≥4

Suggests methods may extend to higher Kronecker powers

Abstract

This note provides a detailed proof of Conner--Gesmundo--Landsberg--Ventura's result that the border rank of the Kronecker square of the little Coppersmith--Winograd tensor is $(q+2)^{2}$.We also indicate how the same ideas seem to extend to the case of the Kronecker cube, pointing toward the conjectural value $(q+2)^{m}$ for $m\ge 4$, although a full proof is left for future work.