Siberian Federal University

Since the beginning of the last century, non-associative algebraic systems have been used to coordinatize non-Desarguesian projective translation planes. Such systems as semifields and quasifields are now find application in cryptographic algorithm design. Hall quasifields were introduced in 1943 and were the first examples of non-distributive and non-associative quasifields. They are two-dimensional quasifields over the center, all non-central elements of which satisfy unique quadratic equation. The automorphism group acts transitively on non-central elements. Hall quasifields of the same order coor- dinatize the isomorphic translation planes, that is Hall planes. Increasing the dimension of the quasifield over the center, we obtain generalized quasifields with the Hall condition. Such a quasifield cannot be either a semifield or a near-field. More than one non-isomorphic quasifields with the Hall condition can correspond to one irreducible polynomial over a given field. The spread set method is used for reasoning and calculations. It allows to present the multiplication rule as a linear transformation and so to de- scribe subfields and sub-quasifields, spectra and automorphisms. These structural questions with some solutions are naturally transferred from the two-dimensional case. Unlike Hall quasifields, a multidimen- sional quasifield with the Hall condition cannot be generated by a single element, this fact completes the results by M. Cordero and V. Jha (2009) on covering and primitivity. Presenting examples, the authors list all non-isomorphic quasifields of order 16 with the Hall condition and define their automorphism groups