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

Mustafa Gezek,Vladimir D Tonchev,Rudi Mathon

doi:10.37236/9008

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.

Highlights

We assume familiarity with the basic facts and notions from design theory, finite geometries, and coding theory [3, 5, 10, 13, 23].the electronic journal of combinatorics 27(1) (2020), #P1.62A 2-(v, k, λ) design is a pair D={X, B} of a set X of v points and a collection B of subsets of X of size k called blocks, such that every two points appear together in exactly λ blocks
The incidence matrix of a design D is a (0, 1)-matrix A = with rows labeled by the blocks, columns labeled by the points, where ai,j = 1 if the ith block contains the jth point, and ai,j = 0 otherwise
If p is a prime, the p-rank of a design D is the rank of its incidence matrix over a finite field of characteristic p

Summary

Introduction

We assume familiarity with the basic facts and notions from design theory, finite geometries, and coding theory [3, 5, 10, 13, 23]. If k > 1, the non-empty intersections of a maximal ((sk − s + 1)k, k)-arc A with lines of a projective plane P of order q = sk are the blocks of a resolvable 2-((sk − s + 1)k, k, 1) design D. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes

Binary codes of designs arising from maximal arcs

Binary codes associated with maximal arcs of degree 4

New connections between projective planes