On Hyperplanes of the Geometry D&lt;sub&gt;4,2&lt;/sub&gt; and their Related Codes

Zayda Zayda

doi:10.3844/jmssp.2009.72.76

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.

Highlights

In[1] Bruyn characterized the hyperplanes of the dual polar spaces DQ (2n,K) and DQ- (2n+1,K)
This study presented a description for two classes of hyperplanes of point-line geometry of type D4,2 which was characterized completely in[7]
In this study we present a new construction for a binary constant weight code using the class of Grassmann geometry A3,2 that are isomorphic to the maximal TI 4-spaces in the classical polar space Ω+(8, F)

Summary

INTRODUCTION

In[1] Bruyn characterized the hyperplanes of the dual polar spaces DQ (2n,K) and DQ- (2n+1,K). A polar space is a point-line geometry Γ = (P, L) satisfying the Buekenhout-Shult axiom: For each point-line pair (p, l) with p not incident with l; p is collinear with one or all points of l, that is p⊥∩l = 1 or else p⊥ ⊃ l. This axiom is equivalent to saying that p⊥ is a geometric hyperplane of Γ for every point p∈ P. A parapolar space is called a strong parapolar space if it has no special pairs

MATERIALS AND METHODS

RESULTS AND DISCUSSION

CONCLUSION