The Tensor Rank of Semifields of Order 16 and 81

TL;DR

This paper computes the tensor rank of all semifields of orders 16 and 81, revealing some have lower multiplicative complexity than the corresponding finite fields, and introduces new code-tensor rank correspondence results.

Contribution

It determines tensor ranks for specific semifields and generalizes a theorem linking linear codes and tensor rank, enhancing computational methods.

Findings

Some semifields of order 81 have lower multiplicative complexity than finite fields.

Generalized theorem of Brockett and Dobkin for arbitrary tensors.

Made tensor rank computation more feasible through new code correspondence.

Abstract

We determine the tensor rank of all semifields of order 16 over $\mathbb{F}_2$ and of all semifields of order 81 over $\mathbb{F}_3$. Our results imply that some semifields of order 81 have lower multiplicative complexity than the finite field $\mathbb{F}_{81}$ over $\mathbb{F}_3$. We prove new results on the correspondence between linear codes and tensor rank, including a generalisation of a theorem of Brockett and Dobkin to arbitrary tensors, which makes the problem computationally feasible.