Multisets Meeting Lines in 0 Modulo p Points and an Improvement of the Bagchi-Inamdar Bound

Abstract

The p-ary code associated with the incidence structure of points and t-spaces in a projective space PG(m,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{{PG}}\,}}(m,q)$$\end{document}, where q=ph\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=p^h$$\end{document}, is the Fp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_p$$\end{document}-subspace generated by the incidence vectors of the blocks of this design. The dual of this code consists of all vectors orthogonal to every codeword of the original code. The minimum weight of this dual code has been studied since the 1970s; the currently best known lower bound is from 2002 by Bagchi and Inamdar. In this paper we provide a substantial improvement of this bound in the case when h>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h>1$$\end{document} and m,p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m, p >2$$\end{document}. The main ingredient of the proof is a lower bound on the size of multisets of AG(m,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{AG}(m,q)$$\end{document} meeting each line in 0modp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \bmod p$$\end{document} points. We prove this, by generalising the polynomial argument in [S. Ball, A. Blokhuis, A. Gács, P. Sziklai, Zs. Weiner: On linear codes whose weights and length have a common divisor, Adv. Math., 211 (2007), 94–104] from (ordinary) planar point sets to multisets in higher dimensional spaces. While the sharpness of the planar result for point sets is in general not known, we can show that our result for multisets is sharp for every h,m,p≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h,m,p\ge 2$$\end{document}.