  The minimum weight of the code generated by the incidence matrix of points
versus lines in a projective plane has been known for over 50 years.
Surprisingly, finding the minimum weight of the dual code of projective planes
of non-prime order is still an open problem, even in the Desarguesian case.
  In this paper, we focus on the case of projective planes of order $p^2$,
where $p$ is prime, and we link the existence of small weight code words in the
dual code to the existence of embedded subplanes and {\em antipodal planes}. In
the Desarguesian case, we can exclude such code words by showing a more general
result that no antipodal plane of order at least 3 can be embedded in a
Desarguesian projective plane.
  Furthermore, we use combinatorial arguments to rule out the existence of code
words in the dual code of points and lines of an arbitrary projective plane of
order $p^2$, $p$ prime, of weight at most $2p^2-2p+4$ using more than two
symbols. In particular, this leads to the result that the dual code of the
Desarguesian projective plane $\mathrm{PG}(2,p^2)$, $p\geq 5$, has minimum
weight at least $2p^2-2p+5$.
