The minimum weight of the code generated by the point-to-line incidence matrix in the projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of the projective plane of non-prime degree is an open problem even for Desarguez.
In this paper, we focus on the case of projective planes of order $p^2$ (where $p$ is prime), and investigate the existence of low-weight code words in dual codes by means of embedded subplanes and {\em vs. plantar surface}. For the Desarguesian, we can rule out such code words by showing the more general result that an antipodal plane of at least degree 3 cannot be embedded in the Desarguesian projective plane.
In addition, the combinatorial argument can be used to find codewords in any projective plane point-line dual code of order $p^2$, $p$ primes, and weights at most $2p^2-2p. Exclude existence. +4$ with 2 or more symbols. In particular, this leads to the dual code of the Desarges projective plane $\mathrm{PG}(2,p^2)$, $p\geq 5$ having a minimum weight of at least $2p^2-2p. increase. +5$.
