Some $p$-ranks related to orthogonal spaces
Journal of Algebraic Combinatorics, Tome 4 (1995) no. 4, pp. 295-316.

Voir la notice de l'article dans Electronic Library of Mathematics

Summary: 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 $P$G$(2, p)$ using secants, or tangents and passants, of a nondegenerate conic.
Mots-clés : $p$-rank, quadric, ovoid, code 
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     title = {Some $p$-ranks related to orthogonal spaces},
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Blokhuis, Aart; Moorhouse, G.Eric. Some $p$-ranks related to orthogonal spaces. Journal of Algebraic Combinatorics, Tome 4 (1995) no. 4, pp. 295-316. https://geodesic-test.mathdoc.fr/item/JAC_1995__4_4_a3/