Some p-Ranks Related to Orthogonal Spaces
Aart Blokhuis
and G.Eric Moorhouse
DOI: 10.1023/A:1022477715988
Abstract
We determine the p-rank of the incidence matrix of hyperplanes of PG( n, p e) and points of a nondegenerate quadric. This yields new bounds for ovoids and the size of caps in finite orthogonal spaces. In particular, we show the nonexistence of ovoids in O 10 + (2 e ), O 10 + (3 e ), O 9 (5 e ), O 12 + (5 e ) O_{10}^ + (2^e ),O_{10}^ + (3^e ),O_9 (5^e ),O_{12}^ + (5^e ) and O 12 + (7 e ) O_{12}^ + (7^e ) . We also give slightly weaker bounds for more general finite classical polar spaces. Another application is the determination of certain explicit bases for the code of PG(2, p) using secants, or tangents and passants, of a nondegenerate conic.
Pages: 295–316
Keywords: $p$-rank; quadric; ovoid; code
Full Text: PDF
References
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2. B. Bagclii and N.S.N. Sastry, "Even order inversive planes, generalized quadrangles and codes," Geom. Ded. 22(1987), 137-147.
3. A.E. Brouwer and H.A. Wilbrink, "Block Designs," in Handbook of Incidence Geometry. Foundations and Buildings, ed. F. Buekenhout, North-Holland, Amsterdam and New York, 1995.
4. P. Dembowski, Finite Geometries, Springer, Berlin and New York, 1968.
5. J.M. Goethals and P. Delsarte, "On a class of majority-logic decodable cyclic codes," IEEE Trans. Inform. Theory 14 (1968), 182-188.
6. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Univ. Press, New York, 1979.
7. J.W.P. Hirschfeld and J.A. Thas, General Galois Geometries, Oxford Univ. Press, Oxford and New York, 1991.
8. W.M. Kantor, "Ovoids and translation planes," Canad. J. Math. 34 (1982), 1195-1207.
9. F.J. MacWilliams and H.B. Mann, "On the /vrank of the design matrix of a difference set," Inform, and Control 12 (1968), 474-489.
10. G.E. Moorhouse, "Bruck nets, codes and characters of loops," Des. Codes and Crypt. 1 (1991), 7-29.
11. E. Shult, "Nonexistenceofovoidsinf2+(10, 3),"y. Comb. Theory, Ser. A 51 (1989), 250-257.
12. K.J.C. Smith, "On the p-rank of the incidence matrix of points and hyperplanes in a finite projective geometry," J. Comb. Theory 1 (1969), 122-129.
13. D.E. Taylor, The Geometry of the Classical Groups, Heldermann Verlag, Berlin, 1992.
14. J.A. Thas, "Ovoids and spreads of finite classical polar spaces," Geom. Ded. 10 (1981), 135-144.