A construction of small complete caps in projective spaces

Abstract

In this work complete caps in P G(N, q) of size O(q N −1 2 log 300 q) are obtained by probabilistic methods. This gives an upper bound asymptotically very close to the trivial lower bound √ 2q N −1 2 and it improves the best known bound in the literature for small complete caps in projective spaces of any dimension. The result obtained in the paper also gives a new upper bound for l(m, 2, q) 4 , that is the minimal length n for which there exists an [n, n − m, 4] q 2 covering code with given m and q.

Small complete caps in ${\rm PG}(4n + 1, q)$

“…r−1 2 . Constructions of complete caps whose size is close to this lower bound are only known for q even.…”

“…A probabilistic approach has been employed in [7] to provide an upper bound on t 2 (r, q). For further or more recent results on this topic see also [2,3,4,9]. For an account on the various constructive methods known so far the reader is referred to [12].…”

On Almost Complete Caps in PG(N, q)

“…These new concepts and the methods of their investigation can be useful for bounds and constructions of small saturating sets and small complete caps, including a rigorous proof of the conjectural bound (4). This paper is organized as follows.…”

Complete Caps in AG(N,q) with BothNandqOdd

“…In all the other cases, all known infinite families of complete caps explicitly constructed have size far from , and even sophisticated probabilistic methods can only ensure the existence of complete caps with no more than points in , with c a constant independent of q ; see for and for . In this case, the probabilistic methods give the same asymptotical bound for complete caps in .…”