Let AR,q denote a family of covering codes, in which the covering radius R and the size q of the underlying Galois field are fixed, while the code length tends to infinity. The construction of families with small asymptotic covering densities is a classical problem in the area Covering Codes.In this paper, infinite sets of families AR,q, where R is fixed but q ranges over an infinite set of prime powers are considered, and the dependence on q of the asymptotic covering densities of AR,q is investigated. It turns out that for the upper limit µ * q (R, AR,q) of the covering density of AR,q, the best possibility is(1)The main achievement of the present paper is the construction of optimal infinite sets of families AR,q, that is, sets of families such that (1) holds, for any covering radius R ≥ 2. We first showed that for a given R, to obtain optimal infinite sets of families it is enough to construct R infinite families A (0) R,q , A (1) R,q , . . . , A (R−1) R,q

More than thirty new upper bounds on the smallest size t2(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 t2(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 m2(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 ≤ m2(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 t2(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 ( 1 2 (q + 3) + δ)-arcs other than conics that share 1 2 (q + 3) points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), q ≡ 2 (mod 3) odd, we propose new constructions of 1 2 (q + 7)-arcs and show that they are complete for q ≤ 3701. Mathematics Subject Classification (2000). 51E21, 51E22, 94B05.

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

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New upper bounds on the smallest size t 2 (2, q) of a complete arc in the projective plane P G(2, q) are obtained for q ≤ 9109. From these new bounds it follows that for q ≤ 2621 and q = 2659, 2663, 2683, 2693, 2753, 2801, the relation t 2 (2, q) < 4.5 √ q holds.Also, for q ≤ 5399 and q = 5413, 5417, 5419, 5441, 5443, 5471, 5483, 5501, 5521, we have t 2 (2, q) < 4.8 √ q. Finally, for q ≤ 9067 it holds that t 2 (2, q) < 5 √ q. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms.

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The spectrum of possible sizes k of complete k-arcs in finite projective planes PG(2, q) is investigated by computer search. Backtracking algorithms that try to construct complete arcs joining the orbits of some subgroup of collineation group P L(3, q) and randomized greedy algorithms are applied. New upper bounds on the smallest size of a complete arc are given for q = 41, 43, 47, 49, 53, 59, 64, 71 ≤ q ≤ 809, q = 529, 625, 729, and q = 821. New lower bounds on the second largest size of a complete arc are given for q = 31, 41, 43, 47, 53, 125. Also, many new sizes of complete arcs are obtained for 31 ≤ q ≤ 167. (2000): 51E21, 51E22, 94B05. Mathematics Subject Classification

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