Journal of Mathematics and Statistics

On Hyperplanes of the Geometry D4,2 and their Related Codes

Abdelsalam Abou Zayda

DOI : 10.3844/jmssp.2009.72.76

Journal of Mathematics and Statistics

Volume 5, Issue 1

Pages 72-76

Abstract

Problem statement: The point-line geometry of type D4,2 was introduced and characterized by many authors such as Shult and Buekenhout and in several researches many of geometries were considered to construct good families of codes and this forced us to present very important substructures in such geometry that are hyperplanes. Approach: We used the isomorphic classical polar space Ω+(8, F) and their combinatorics to construct the hyperplanes and the family of certain codes related to such hyperplanes. Results: We proved that each hyperplane is either the set Δ2 (p) which consisted of all points at a distance mostly 2 from a fixed point p or a Grassmann geometry of type A3,2 and then we presented a new family of non linear binary constant-weight codes. Conclusion: The hyperplanes of the geometry D4,2 allow us to discuss further substructures of the geometry such as veldkamp spaces.

Copyright

© 2009 Abdelsalam Abou Zayda. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.