Some Results on Quadrics in Finite Projective Geometry Based on Galois Fields

D. K. Ray-Chaudhuri

doi:10.4153/CJM-1962-010-2

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a paper (5) published in the Proceedings of the Cambridge Philosophical Society, Primrose obtained the formulae for the number of points contained in a non-degenerate quadric in PG(n, s), the finite projective geometry of n dimensions based on a Galois field GF(s). In § 3 of the present paper the formulae for the number of p-flats contained in a non-degenerate quadric in PG(n, s) are obtained. In § 4 an interesting property of a non-degenerate quadric in PG(2k, 2m) is proved. These properties of a quadric will be used in solving some combinatorial problems of statistical interest in a later paper.

References

1. Barlotti, Adriano, Un1 estensione del teorema di Segre-Kustaanheimo, Boll. Un. Mat. Ital., 10 (1955), 498–506.Google Scholar

2. Barlotti, Adriano, Un'osservazione sulle k-calotte degli shazi lineari finiti di dirnensione tre. Boll. Un. Mat. Ital, 11 (1956), 248–252.Google Scholar

3. Bose, R. C., Mathematical theory of the symmetrical factorial design, Sankhya, 8 (1947), 107–166.Google Scholar

4. Dickson, L. E., Linear groups, Teubner (1901), 197–198.Google Scholar

5. Primrose, E. J. F., Quadrics in finite geometries, Proc. Camb. Phil. Soc, 4-7 (1951), 299–304.Google Scholar

6. Qvist, B., Some remarks concerning curves of the second degree in a finite plane, Ann. Acad. Sci. Fenn., 134 (1952).Google Scholar

7. Segre, B., Curve razionali e k-archi negli spazi finiti, Ann. Mat. Pura Appl. (4), 39 (1955), 357–379.Google Scholar

8. Tallini, Giuseppe, Suite k-calotte degli spazi Lineari finiti, Alti. Accad. Naz. Linceai. Rend. Cl. Sci. Fis. Mat. Nat., 20 (1956), 311–317.Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Ray-Chaudhuri, D. K. 1965. Some Configurations in Finite Projective Spaces and Partially Balanced Incomplete Block Designs. Canadian Journal of Mathematics, Vol. 17, Issue. , p. 114.

BOSE, R.C. 1965. Contributions to Statistics. p. 21.

Bose, R. C. and Chakravarti, I. M. 1966. Hermitian Varieties in a Finite Projective Space PG(N, q2). Canadian Journal of Mathematics, Vol. 18, Issue. , p. 1161.

Ray-Chaudhuri, D. K. 1968. Combinatorial Information Retrieval Systems for Files. SIAM Journal on Applied Mathematics, Vol. 16, Issue. 5, p. 973.

Ghosh, Sakti P. 1970. On Quadrics in Finite Euclidean Geometries and Application to Construction of Balanced Incomplete Block Designs. SIAM Journal on Applied Mathematics, Vol. 19, Issue. 4, p. 730.

Dye, R.H 1971. On the conjugacy classes of involutions of the simple orthogonal groups over perfect fields of characteristic two. Journal of Algebra, Vol. 18, Issue. 3, p. 414.

Bose, R. C. and Shrikhande, S. S. 1972. Geometric and pseudo-geometric graphs (q2+1,q+1,1). Journal of Geometry, Vol. 2, Issue. 1, p. 75.

Hubaut, Xavier L. 1975. Strongly regular graphs. Discrete Mathematics, Vol. 13, Issue. 4, p. 357.

Wolfmann, J. 1975. Codes projectifs a deux ou trois poids associfs aux hyperquadriques d'une geometrie finie. Discrete Mathematics, Vol. 13, Issue. 2, p. 185.

Vecchi, Italo 1984. Some results on coverings of galois spaces with ovoids and related bib designs. Journal of Statistical Planning and Inference, Vol. 10, Issue. 2, p. 219.

Games, Richard A. 1986. The Geometry of m-Sequences: Three-Valued Crosscorrelations and Quadrics in Finite Projective Geometry. SIAM Journal on Algebraic Discrete Methods, Vol. 7, Issue. 1, p. 43.

Games, R. 1986. The geometry of quadrics and correlations of sequences (Corresp.). IEEE Transactions on Information Theory, Vol. 32, Issue. 3, p. 423.

Casaglia, Ivan 1989. Some block designs associated with quadrics of PG(4, q) and PG(5, q. Journal of Statistical Planning and Inference, Vol. 21, Issue. 2, p. 265.

Yang, Benfu and Wei, Wandi 1990. Finite orthogonal geometries with characteristic ≠ 2 and PBIB designs. I. Linear Algebra and its Applications, Vol. 134, Issue. , p. 1.

Chakravarti, I. M. 1990. Coding Theory and Design Theory. Vol. 20, Issue. , p. 35.

Aubry, Yves 1992. Coding Theory and Algebraic Geometry. Vol. 1518, Issue. , p. 4.

Wan, Zhe-xian 1996. Group Theory in China. p. 182.

Wan, Zhe-xian 1997. Geometry of classical groups over finite fields and its applications. Discrete Mathematics, Vol. 174, Issue. 1-3, p. 365.

Suogang, Gao and Yangxian, Wang 1999. Geometric construction of some families of two-class and three-class association schemes from nondegenerate quadrics in characteristic two. Acta Mathematicae Applicatae Sinica, Vol. 15, Issue. 1, p. 24.

Wang, Kaishun and Wei, Hongzeng 2000. GEOMETRIC CONSTRUCTION OF ASSOCIATION SCHEMES FROM DEGENERATE QUADRICS IN PROJECTIVE SPACES PG (2 v + δ + l − 1, F q ) FOR q ODD. Acta Mathematica Scientia, Vol. 20, Issue. 1, p. 35.

Download full list

Article contents

Some Results on Quadrics in Finite Projective Geometry Based on Galois Fields

Extract

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests