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Projective Geometries that are Disjoint Unions of Caps

Published online by Cambridge University Press:  20 November 2018

Barbu C. Kestenband
Barbu C. Kestenband*
Affiliation:
New York Institute of Technology, Old Westbury, New York
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We show that any PG(2n, q2) is a disjoint union of (q2n+1 − 1)/ (q − 1) caps, each cap consisting of (q2n+1 + 1)/(q + 1) points. Furthermore, these caps constitute the “large points” of a PG(2n, q), with the incidence relation defined in a natural way.

A square matrix H = (hij) over the finite field GF(q2), q a prime power, is said to be Hermitian if hijq = hij for all i, j [1, p. 1161]. In particular, hii ∈ GF(q). If if is Hermitian, so is p(H), where p(x) is any polynomial with coefficients in GF(q).

Given a Desarguesian Projective Geometry PG(2n, q2), n > 0, we denote its points by column vectors:

All Hermitian matrices in this paper will be 2n + 1 by 2n + 1, n > 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1299 - 1305
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Bose, R. C. and Chakravarti, I. M., Hermitian varieties in a finite projective space PG(N, q2), Can. J. Math. 17 (1966), 1161–1182.Google Scholar
2. Hall, M. Jr., Combinatorial theory (Blaisdell, 1967).Google Scholar
3. Singer, J., A theorem infinite projective geometry and some applications to number theory, Trans. Amer. Math. Soc. 43 (1938), 377–385.Google Scholar