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Volume 21 (2025) Article 13 pp. 1-17
Towards Finding Hay in a Haystack: Explicit Tensors of Border Rank Greater Than $2.02m$ in $\mathbb{C}^m\otimes \mathbb{C}^m\otimes \mathbb{C}^m$
Received: May 11, 2021
Revised: September 3, 2025
Published: December 25, 2025
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Keywords: matrix multiplication complexity, tensor rank, asymptotic rank, laser method
Categories: complexity, matrix multiplication, tensor rank, asymptotic rank, laser method, short
ACM Classification: F.1.3
AMS Classification: 15A69, 14L35, 68Q15

Abstract: [Plain Text Version]

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We write down an explicit sequence of tensors in $\mathbb{C}^m\otimes \mathbb{C}^m\otimes \mathbb{C}^m$, 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.