Explicit tensors exceeding the 3n border-rank threshold

Construct explicit tensors of border rank, or tensor rank, greater than 3n using techniques beyond border substitution combined with linear flattenings on restricted tensors.

Background

The paper constructs polynomial-time explicit n × n × n tensors with border rank at least 3n − o(n), improving the previous explicit lower bound. It also explains that the construction relies on border substitution and Koszul, hence linear, flattenings applied to restricted tensors.

The authors explicitly ask whether the 3n threshold can be surpassed and identify border apolarity and Kronecker–Young flattenings as possible methods capable of going beyond the techniques used in the paper. The question concerns both border rank and, potentially, the stronger tensor-rank measure.

References

Can we find explicit tensors of border rank (or even, tensor rank) greater than $3n$? This will require techniques beyond the method of border substitution + linear flattenings on the restricted tensors, such as border apolarity or the recently introduced Kronecker-Young flattenings.

— Explicit tensors beyond the linear flattening barrier  (2610.08504 - Lovitz, 6 Oct 2026) in Section 1, subsection “Open questions,” item 1

To what extent can these techniques be generalized to exhibit explicit non-square tensors beyond the linear flattening barrier?

— Explicit tensors beyond the linear flattening barrier  (2610.08504 - Lovitz, 6 Oct 2026) in Section 1, subsection “Open questions,” item 3