Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order $p^2$

TL;DR

This paper investigates the minimum weight of the dual code of points and lines in projective planes of order p^2, linking small code words to embedded antipodal planes and establishing lower bounds for the Desarguesian case.

Contribution

It establishes new bounds on the minimum weight of the dual code in projective planes of order p^2 and relates code words to embedded antipodal planes, extending understanding in non-prime order cases.

Findings

No antipodal plane of order ≥ 3 can be embedded in a Desarguesian projective plane.

Dual code of PG(2,p^2) for p ≥ 5 has minimum weight at least 2p^2 - 2p + 5.

Bound of at most 2p^2 - 2p + 4 for code words with more than two symbols.

Abstract

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 $2p^2-2p+5$.