Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932 Note Mat. 33 (2013) no. 2, 43–64. doi:10.1285/i15900932v33n2p43 Classification of book spreads in PG(5, 2) T.P. McDonough Institute of Mathematics and Physics, Aberystwyth University, UK tpd@aber.ac.uk R. Shaw Centre for Mathematics, University of Hull, UK r.shaw@hull.ac.uk S. Topalova i Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria svetlana@math.bas.bg Received: 6.4.2013; accepted: 2.5.2013. Abstract. We classify all line spreads S21 in PG(5, 2) of a special kind, namely those which are book spreads. We show that up to isomorphism there are precisely nine different kinds of book spreads and describe the automorphism groups which stabilize them. Most of the main results are obtained in two independent ways, namely theoretically and by computer. Keywords: line spread, PG(5, 2), combinatorial design MSC 2010 classification: 51E20, 05B25, 05B05 1 Introduction A line spread in the projective space PG(5,q) consists of a set of q 4 + q 2 +1 lines which partition the points of the space. The task of classifying all line spreads in PG(5,q) is an extremely formidable one, and certainly requires com- puter help even for low values of q. Line spreads in PG(5, 2) were considered in [13], where, with computer assistance, 131044 inequivalent spreads were found. Most of these spreads have very little symmetry and presumably their prop- erties do not warrant further consideration. Indeed, see [13, Table I], as many as 128474 different kinds of line spreads in PG(5, 2) have trivial automorphism group! In this paper we classify all line spreads S 21 in PG(5, 2) of a special kind, namely those which are book spreads. We claim that book spreads are some of the most interesting kinds of spreads. For since their lines partition not only the whole projective space, but also subspaces covering it, they have rich automor- i This work is partially supported by the Bulgarian National Science Fund under Contract No I01/0003. http://siba-ese.unisalento.it/ c  2013 Universit`a del Salento brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by ESE - Salento University Publishing