Determination of division algebras with 243 elements

Finite nonassociative division algebras (i.e., finite semifields) with 243 elements are completely classified.Comment: 6 pages, 3 table

Inner Automorphisms of Finite Semifields

EnUnlike finite fields, finite semifields possess inner automorphisms. A further surprise is that even noncommutative semifields possess inner automorphisms. We compute inner automorphisms and automorphism groups for semifields quadratic over the nucleus, the Hughes-Kleinfeld semifields and the Dickson commutative semifields

Finite semifields and nonsingular tensors

In this article, we give an overview of the classification results in the theory of finite semifields (note that this is not intended as a survey of finite semifields including a complete state of the art (see also Remark 1.10)) and elaborate on the approach using nonsingular tensors based on Liebler (Geom Dedicata 11(4):455-464, 1981)

Division, adjoints, and dualities of bilinear maps

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the adjoint category is not a module category but nevertheless it is suitably familiar. The universal properties have geometric perspectives. For example, products are orthogonal sums. The bilinear division maps are the simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes the understanding that the atoms of linear geometries are algebraic objects with no zero-divisors. Adjoint-isomorphism coincides with principal isotopism; hence, nonassociative division rings can be studied within this framework. This also corrects an error in an earlier pre-print; see Remark 2.11

A historical perspective of the theory of isotopisms

In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.Junta de Andalucí

Automorphisms and isomorphisms of Jha-Johnson semifields obtained from skew polynomial rings

We study the automorphisms of Jha-Johnson semifields obtained from a right invariant irreducible twisted polynomial f Є K[t;σ], where K = Fqn is a finite field and σ an automorphism of K of order n, with a particular emphasis on inner automorphisms and the automorphisms of Sandler and Hughes-Kleinfeld semifields. We include the automorphisms of some Knuth semifields (which do not arise from skew polynomial rings). Isomorphism between Jha-Johnson semifields are considered as well

Presemifields, bundles and polynomials over GF (pn)

The content of this thesis is first and foremost about presemifields and the equivalence classes they may be categorized by. This equivalence has been termed “bundle equivalence'' by Horadam. Bundle equivalence is inherited from multiplicative orthogonal cocycles, and the final Chapter is devoted entirely to coboundaries and cocycles. In this thesis we provide a complete computational classification of the bundles of presemifields in all presemifield isotopism classes of order p n , provide a formula for the number of bundles in the presemifields isotopism class of GF (p 2 ) and give a representative of each bundle, for any prime p . We provide computational classification of the bundles of presemifields in the isotopism class of GF (p 3 )  for the cases  p =3,5,7,11 and give representatives, give formulae for two of the three possible size bundles in the presemifield isotopism class of  GF (p 3 )   which we call the minimum and the mid-size bundles. We provide a Conjecture which states the total number of mid-size bundles in the isotopism class of  GF (p 3 ) and give a computational classification of the bundles of presemifields in the isotopism class of  GF (2 5 ) and  GF (3 4 ) . We provide a measurement of the differential uniformity of functions derived from the diagonal map of presemifield multiplications with order p n < 16 and derive bivariate polynomial formulae for cocycles and coboundaries in We produce a basis for the ( p n - 1 - n ) - dimensional -space of coboundaries. When p = 2 we give a recursive definition of the basis coboundaries. We use the Kronecker product to explain the self-similarity of the binomial coefficients modulo a prime and use the Kronecker product to define recursively the basis coboundaries for p odd, and we demonstrate this holds for the case p = 2. We show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form when  p = 2.  The results of this thesis have been published in the Proceedings of the International Workshop on Coding and Cryptography, Designs, Codes and Cryptography and the Proceedings of IEEE International Symposium on Information Theory and will appear in the Journal of the Australian Mathematical Society