On translation spreads of H(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(q)$$\end{document}

Using the connection between translation spreads of H(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(q)$$\end{document} and linear sets (see Cardinali et al. in Eur J Comb 23:367–376, 2002; Lunardon and Polverino in J Algebraic Comb 18:255–262, 2003), some results on the existence of translation spreads of H(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(q)$$\end{document} are given, improving classification results contained in Offer (Spreads and ovoids of the split Cayley hexagon, 2000) and in Bonoli and Polverino (Discrete Math 296:129–142, 2005).