New advances in the computational exploration of semifields

Abstract

Finite semifields (finite non-necessarily associative division rings) have traditionally been considered in the context of finite geometries (they coordinatize projective semifield planes). New applications to the coding theory, combinatorics and the graph theory have broadened the potential interest in these rings. We show recent progress in the study of these objects with the help of computational tools. In particular, we state results on the classification and primitivity of semifields obtained with the help of advanced and efficient implementations (both sequential and parallel) of different algorithms specially designed to manipulate these objects.

Keywords:

• finite semifield
• finite division ring
• computational methods
• classification
• primitivity

2010 AMS Subject Classifications :

• 12K10
• 17D99
• 17-04
• 17-08
• 51A35

Acknowledgements

The authors thankfully acknowledge the computer resources, technical expertise and assistance provided by the Centro de Supercomputación y Visualización de Madrid (CeSViMa) and the Spanish Supercomputing Network. E.F. Combarro and J. Ranilla are partially supported by MICINN-TIN2010-14971, MEC-TIN2007-61273 and MEC-TIN2007-29664E and I.F. Rúa is partially supported by MEC-MTM-2010-18370-C04-01, MEC-MTM2007-67884 C04-01 and IB08-147.