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We determine the border ranks of tensors that could potentially advance theknown upper bound for the exponent $\omega$ of matrix multiplication. TheKronecker square of the small $q=2$ Coppersmith-Winograd tensor equals the$3\times 3$ permanent, and could potentially be used to show $\omega=2$. Weprove the negative result for complexity theory that its border rank is $16$,resolving a longstanding problem. Regarding its $q=4$ skew cousin in $C^5\otimes C^5\otimes C^5$, which could potentially be used to prove$\omega\leq 2.11$, we show the border rank of its Kronecker square is at most$42$, a remarkable sub-multiplicativity result, as the square of its borderrank is $64$. We also determine moduli spaces $\underline{VSP}$ for the smallCoppersmith-Winograd tensors.

Bad and good news for Strassen's laser method: Border rank of the 3x3 permanent and strict submultiplicativity