ISSN:
1572-9168
Keywords:
projective space
;
quadrics
;
complete caps
;
asymptotic maximal size of caps.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract A cap of a quadric is a set of its points whose pairwise joins are all chords. Such a cap is complete if it is not part of a larger one. Few examples of complete caps are known except for quadrics in low dimensions. In this paper, we consider the case when the coordinate field is GF(p), with p an odd prime, and construct, in each projective space GF(n,p) with n ≥ p − 1 and n ≢ − 2(mod p), a cap on one of its nonsingular quadrics. We use this in two ways. Firstly, we combine its size with the recent Blokhuis–Moorhouse upper bound for quadric caps to show that the size of the largest cap of any nonsingular quadric in PG(N,p) is asymptotic to Np − 1/(p − 1) ! as N tends to infinity. Secondly, by establishing situations when our cap is complete, we produce various infinite families of complete quadric caps over GF(p) for each p. Earlier work determined all complete caps of all nonsingular quadrics over GF(2).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1005039300405
Permalink