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
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
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
