Maximal arcs, codes, and new links between projective planes of order 16
Document Type
Article
Publication Date
3-20-2020
Department
Department of Mathematical Sciences
Abstract
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 P G(2,2m) is formulated.
Publication Title
The Electronic Journal of Combinatorics
Recommended Citation
Gezek, M.,
Mathon, R.,
&
Tonchev, V.
(2020).
Maximal arcs, codes, and new links between projective planes of order 16.
The Electronic Journal of Combinatorics,
27(1).
http://doi.org/10.37236/9008
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/1766
Publisher's Statement
©The authors. Released under the CC BY-ND license (International 4.0).