The Tensor Rank of Semifields of Order 16 and 81

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.