A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the number of required base field multiplications is the tensor rank, or the multiplicative complexity. The other base field operations are additions and scalings by constants, which together we refer to as the additive complexity. When used recursively, the tensor rank determines the exponent while the other operations determine the constant of the associated asymptotic complexity bounds. For small extensions, both measures are of similar importance.

Abstract. By exploiting the connection between scattered [Formula: see text]-subspaces of [Formula: see text] and minimal nondegenerate 3-dimensional rank-metric codes of [Formula: see text], [Formula: see text], described in [ G. N. Alfarano et al., J. Combin. Theory Ser. A, 192 (2022), 105658 ], we will exhibit a new class of codes with parameters [Formula: see text] for infinite values of [Formula: see text] and [Formula: see text] odd. Moreover, by studying the geometric structures of these scattered subspaces, we determine the rank weight distribution of the associated codes.