Towards finding hay in a haystack: explicit tensors of border rank greater than $2.02m$ in $C^m\otimes C^m\otimes C^m$

Abstract

We write down an explicit sequence of tensors in $C^m\otimes C^m\otimes C^m$, for all $m$ sufficiently large, having border rank at least $2.02m$, overcoming a longstanding barrier. We obtain our lower bounds via the border substitution method.