Coding Theory and Design Theory pp 1-8 | Cite as

# Baer Subplanes, Ovals and Unitals

## Abstract

1. Introduction. In this paper we examine some of the types of codewords that can occur in codes associated with finite projective and affine planes, exploring further the notions that were introduced in [1]. There we defined the hull, \(
H_p \left( D \right)
\), of a design \(
\left( D \right)
\) over a finite field \(F_p\), where *p* is a prime that divides the order *n* of the design: if \(
C_p \left( D \right)
\) denotes the code of \(
\left( D \right)
\) over *F* _{ p }, defined to be the space spanned by the characteristic functions of the blocks of \(
\left( D \right)
\), then \(
H_p \left( D \right)\; = \;C_p \left( D \right)\; \cap \;C_P \left( D \right)^ \bot
\), where *X* ^{⊥} denotes the subspace orthogonal to *X* with respect to the standard inner product. In the case where \(
\left( D \right)
\) is a finite projective plane П, and π any affine part of П, we showed that affine planes obtained by “derivation” from π (see [1]) could be obtained from minimal-weight vectors of \(H_p(\pi)^{\bot}\). Since \( {{H}_{p}}{{(\pi )}^{ \bot }} \) is the image of the natural projection of \(H_p(\Pi)^{\bot}\) we were led to examine the nature of codewords of the code \(H_p(\Pi)^{\bot}\), and, in particular, those codewords that could give rise to minimal-weight (i.e. weight-*n*) vectors in \(H_p(\pi)^{\bot}\) and \(C_p(\pi)\). The support of such a vector must form either a line or a blocking set for П; examples of these, other than lines, are Bær subplanes meeting the line at infinity for π in a line segment, or, in the even-order case, ovals meeting that line in two points.

### Keywords

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### References

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