Plane (s,3)-Arcs in PG(2,b)
(s, r)-caps are sets of s points in some space such that not more than r points are collinear. In this terminology normal s-caps are (s, 2)-caps. (s, r)-caps in the projective plane PG(2, b) over Fb are also known as plane (s, r)-arcs. (s, 3)-arcs in PG(2, b) correspond directly to linear [s, 3, s−3]-codes over Fb.
In [1] the following (s, 3)-arcs in PG(2, b) are shown to exist:
| b | s | Resulting code |
| 3 | 9 | [9, 3, 6]-code over F3 |
| 4 | 9 | [9, 3, 6]-code over F4 |
| 5 | 11 | [11, 3, 8]-code over F5 |
| 7 | 15 | [15, 3, 12]-code over F7 |
| 8 | 15 | [15, 3, 12]-code over F8 |
| 9 | 17 | [17, 3, 14]-code over F9 |
| 11 | 21 | [21, 3, 18]-code over F11 |
| 13 | 23 | [23, 3, 20]-code over F13 |
References
| [1] | Simeon Ball. On Sets of Points in Finite Planes. PhD thesis, University of Sussex, U.K., 1994. |
Copyright
Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Plane (s,3)-Arcs in PG(2,b).”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2024-09-05.
http://mint.sbg.ac.at/desc_CAMDSCodeN3.html