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

Sam Adriaensen, Tamás Szőnyi, Zsuzsa Weiner

Published 2024-11-28Version 1

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 $5p-36$.