Maximal arcs in PG(2, q) and partial flocks of the quadratic cone

Abstract The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52,4,1) designs arising from previously known maximal (52,4)-arcs, as well as some newly discovered maximal (52,4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(120,8,1) design associated with a maximal (120,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose–Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52,4,1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16.

Download Full-text

Some maximal arcs in Hall planes

Journal of Geometry ◽

10.1007/bf01406830 ◽

1995 ◽

Vol 52 (1-2) ◽

pp. 101-107 ◽

Cited By ~ 5

Author(s):

Nicholas Hamilton

Keyword(s):

Maximal Arcs

Download Full-text

Maximal Arcs, Codes, and New Links Between Projective Planes of Order 16

The Electronic Journal of Combinatorics ◽

10.37236/9008 ◽

2020 ◽

Vol 27 (1) ◽

Cited By ~ 1

Author(s):

Mustafa Gezek ◽

Rudi Mathon ◽

Vladimir D. Tonchev

Keyword(s):

Minimum Distance ◽

Linear Codes ◽

Projective Planes ◽

Upper And Lower Bounds ◽

Dual Codes ◽

Binary Linear Codes ◽

Incidence Matrices ◽

Maximal Arc ◽

Maximal Arcs ◽

Even Order

In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The binary linear codes of length 52 spanned by the incidence matrices of 2-$(52,4,1)$ designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in $PG(2,2^m)$ is formulated.

Download Full-text