A365198 Smallest k such that there exists a complete k-arc on the projective plane over GF(q), where q = A246655(n) is the n-th prime power > 1.

4, 4, 6, 6, 6, 6, 6, 7, 8, 9, 10, 10, 10, 12, 12, 13, 14, 14

Offset: 1

Author

Robin Visser, Aug 26 2023

Keywords

• nonn
• hard
• more

Comments

A k-arc is a set of k points in PG(2,q) (the projective plane over GF(q)) such that no three are collinear. A complete k-arc is a k-arc which is not contained in any (k+1)-arc.

References

• J. W. P. Hirschfeld, Projective geometries over finite fields, Second edition, Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York, 1998.

Links

• D. Bartoli, G. Faina, S. Marcugini and F. Pambianco, On the minimum size of complete arcs and minimal saturating sets in projective planes, J. Geom. 104 (2013), no. 3, 409-419.
• S. Marcugini, A. Milani, and F. Pambianco, Minimal complete arcs in PG(2,q), q <= 32, arXiv:1005.3412 [math.CO], 2010.
• B. Segre, Le geometrie di Galois, Ann. Mat. Pura Appl. (4) 48 (1959), 1-96.

Crossrefs

Cf. A365216.

Formula

a(n) > sqrt(2*A246655(n)) + 1 [Segre].