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

Abstract

More than thirty new upper bounds on the smallest size t 2(2, q) of a complete arc in the plane PG(2, q) are obtained for (169 ≤ q ≤ 839. New upper bounds on the smallest size t 2(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m 2(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m 2(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t 2(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete $${(\frac{1}{2}(q + 3) + \delta)}$$ -arcs other than conics that share $${\frac{1}{2}(q + 3)}$$ points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), $${{q \equiv 2}}$$ (mod 3) odd, we propose new constructions of $${\frac{1}{2} (q + 7)}$$ -arcs and show that they are complete for q ≤ 3701.