Complete caps in projective spaces PG (n, q)

Abstract

A computer search in the finite projective spaces PG(n, q) for the spectrum of possible sizes k of complete k-caps is done. Randomized greedy algorithms are applied. New upper bounds on the smallest size of a complete cap are given for many values of n and q. Many new sizes of complete caps are obtained. (2000): 51E21, 51E22, 94B05. Mathematics Subject Classification

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

“…With the help of the randomized greedy algorithms [17,23], we obtained more than thirty small complete arcs in PG (2, q) giving new upper bounds on t 2 (2, q). The updated table of t 2 (2, q) is given as Table 1.…”

“…This leads to updated tables of upper bounds for t 2 (n, q), n ≥ 2, and for the spectrum of known sizes of complete caps; see [17,23]. In particular, new upper bounds on t 2 (n, q) for values of n and q are obtained as follows: 5 5, 7, 8, 9 6 4, 8 The new lower bound 534 ≤ m 2 (8, 3) is given.…”

“…For new computer results we used the randomized greedy algorithms considered in [17,Section 2], [23,Section 2], the back-tracking algorithms [17,Section 2], the breadth-first algorithm [65,Section 3], algorithms combining orbits of groups, and other geometrical algorithms.…”

Small complete caps inPG(N,q),qeven

“…Interestingly, this problem is related to Coding Theory. In fact, complete k-caps in PG(N, q) with k > N + 1 and linear quasi-perfect [k, k − N − 1, 4]-codes over F q are equivalent objects (with the exceptions of the complete 5-cap in PG (3,2) giving rise to a binary [5,1,5]-code, and the complete 11-cap in PG (4,3) corresponding to the Golay [11,6,5]-code over F 3 ), see for example [12].…”

“…Their size k equals 2 N/2 + 2 (N+2)/2 − 3 and almost attains the upper bound in (1.1). For q > 2, q even, apart from small values of both N and q (see [1,6,7,16,19]), the best upper bound on t 2 (N, q), N > 3, is due to Pambianco and Storme [21], who proved that For N = 3, t 2 (3, q) ≤ 3q + 2 was first proved in [23], whereas in [22], it is shown that t 2 (3, q) ≤ 2q + t 2 (2, q) (see also [16, Table 4.8]).…”