The Tensor Rank of Semifields of Order 16 and 81

We determine the tensor rank of all semifields of order 16 over F2 and of all semifields of order 81 over F3. Our results imply that some semifields of order 81 have lower multiplicative complexity than the finite field F81 over F3. 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.

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