Maximal arcs, codes, and new links between projective planes

TL;DR

This paper explores the relationship between maximal arcs in projective planes and binary linear codes, deriving bounds, analyzing specific codes of length 52, and revealing new links between different projective planes.

Contribution

It introduces new bounds on code parameters, classifies codes from maximal arcs in order 16 planes, and establishes novel connections between different projective planes.

Findings

Bounds on the 2-rank of incidence matrices are established.

Codes from maximal arcs in order 16 planes are classified up to equivalence.

New links between nonisomorphic projective planes are demonstrated.

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 $PG(2,2^m)$ is formulated.