Recursive constructions of complete caps

Abstract

We present three constructions of complete caps in PG(d; q), q odd, where complete caps in a projective space of smaller dimension are involved. We thereby obtain new series of upper bounds on n2(d; q), the smallest number of points in a complete cap in PG(d; q). The constructions show that for k ? 0, n2(k +1 ; 3) 6 2n2(k; 3); n2(4k +2 ;q ) 6 q 2k+1 + n2(2k; q) for q ? 5 an odd prime power; and n2(4k +2 ;q ) 6 q 2k+1 − (q +1 ) +n2(2k; q )+ n2(2 ;q ) for q ? 9 an odd prime power. c � 2001 Elsevier Science B.V. All rights reserved. MSC: 51E22