  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.
