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.

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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

We give a geometric interpretation of additive quantum stabilizer codes in terms of sets of lines in binary symplectic space. It is used to obtain synthetic geometric constructions and non-existence results. In particular several open problems are removed from Grassl's database [13].

Abstract-The best parameters of quaternary additive codes of small length are determined using the geometric description. Only one open question remains for length ≤ 13. Among the results obtained in this work are the non-existence of [12, 7, 5]-codes and [12, 4.5, 7]-codes as well as the existence of a [13, 7.5, 5]−code.

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