Rank classification of tensors over

Abstract

We consider tensors of format over the finite field . We use computer algebra to classify these tensors by their tensor rank, thus determining the maximum tensor rank to be 9. As a corollary, we provide a new upper bound that the maximum rank of an order-n tensor of format , for , over is at most . We also determine that there are 261 canonical forms of the rank 9 (maximum rank) tensors under the action of , the semi-direct product of (a direct product of) general linear groups with the symmetric group on five elements.