Multisets with few special directions and small weight codewords in Desarguesian planes

TL;DR

This paper explores the relationship between special directions in affine planes and codewords in projective plane codes, providing bounds and classifications for codewords supported on concurrent lines.

Contribution

It establishes a connection between multiset directions and codewords, offering bounds on special directions and extending codeword classifications in projective plane codes.

Findings

Multisets with few special directions relate to codewords supported on concurrent lines.

Lower bounds on the number of special directions are derived using projection function bounds.

Extended classification of codewords supported on multiple concurrent lines for large prime orders.

Abstract

In this paper, we tie together two well studied topics related to finite Desarguesian affine and projective planes. The first topic concerns directions determined by a set, or even a multiset, of points in an affine plane. The second topic concerns the linear code generated by the incidence matrix of a projective plane. We show how a multiset determining only $k$ special directions, in a modular sense, gives rise to a codeword whose support can be covered by $k$ concurrent lines. The reverse operation of going from a codeword to a multiset of points is trickier, but we describe a possible strategy and show some fruitful applications. Given a multiset of affine points, we use a bound on the degree of its so-called projection function to yield lower bounds on the number of special directions, both in an ordinary and in a modular sense. In the codes related to projective planes of prime order $p$, there exists an odd codeword, whose support is covered by 3 concurrent lines, but which is not a linear combination of these 3 lines. We generalise this codeword to codewords whose support is contained in an arbitrary number of concurrent lines. In case $p$ is large enough, this allows us to extend the classification of codewords from weight at most $4p-22$ to weight at most $5p-36$.