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

被引：18
作者：
Davydov A.A. [1 ]
Faina G. [2 ]
Marcugini S. [2 ]
Pambianco F. [2 ]
机构：
[1] Institute for Information Transmission Problems, Russian Academy of Science, Moscow 127994, Bol'shoi Karetnyi per. 19
[2] Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Perugia 06123
来源：
Journal of Geometry | 2009年 / 94卷 / 1-2期
关键词：
Complete arcs in planes; Complete caps; Computer search; Projective spaces; Small complete caps and arcs;
D O I：
10.1007/s00022-009-0009-3
中图分类号：
学科分类号：
摘要：
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, 2h) 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 -arcs other than conics that share points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), (mod 3) odd, we propose new constructions of -arcs and show that they are complete for q ≤ 3701. © 2009 Birkhäuser Verlag Basel/Switzerland.
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页码：31 / 58
页数：27
相关论文
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