  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.
