The maximum rank of 2 × ⋯ × 2 tensors over 𝔽₂

Abstract

We determine that the maximum rank of an order-n $(\ge 2)$ tensor with format $2\times \cdots \times 2$ over the finite field ${\mathbb{F}}_2$ is $2\cdot 3^{n/2-1}$ for even n, and $3^{\lfloor n/2\rfloor }$ for odd n. Since tensor rank is non-increasing upon taking field extensions, ${\mathbb{F}}_2$ gives the largest rank attainable for this tensor format. We also determine a maximum rank canonical form and compute its orbit under the action of the symmetry group $G{L}_2({\mathbb{F}}_2{)}^{\times n}$, and prove that this is the unique maximum rank canonical form, for even $n\ge 2$.

Keywords:

• Multidimensional arrays
• tensor rank
• finite fields
• computer algebra

2010 Mathematics Subject Classifications:

• Primary: 15A21. Secondary: 15-04
• 15A69
• 15B33
• 20B25
• 20C20
• 20G40

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is funded by a scholarship from SSHRC.