caa a22 4500 475807634 CHVBK 20180406123751.0 cr unu---uuuuu 170329e20001001xx s 000 0 eng 10.1023/A:1008304115031 doi (NATIONALLICENCE)springer-10.1023/A:1008304115031 Shaw Ron Department of Mathematics, University of Hull, HU67RX, Hull, United Kingdom aut Subsets of PG(n,2) and Maximal Partial Spreads in PG(4,2) [Elektronische Daten] [Ron Shaw] Put θ n = # {points in PG(n,2)} and φ n = #{lines in PG(n,2)}. Let ψ be anypoint-subset of PG(n,2). It is shown thatthe sum of L = #{internal lines of ψ} and L′= #{external lines of ψ} is the same for all ψ having the same cardinality:[6pt] Theorem A If k is defined by k = |ψ| − θ n − 1, then $$L + L' = \phi _{n - 1} + k(k - 1)/2.$$ (The generalization of this to subsets of PG(n,3) is also obtained.) Let $$\mathcal{S}$$ be a partial spreadof lines in PG(4,2) and let N denote the number of reguli contained in $$\mathcal{S}$$ .Use of Theorem A gives rise to a simple proof of:[6pt] Theorem B If $$\mathcal{S}$$ is maximal then one of the followingholds: (i) $$\left| \mathcal{S} \right| = 5,{\text{ }}N = 10;{\text{ }}$$ (ii) $$\left| \mathcal{S} \right| = 7,{\text{ }}N = 4;{\text{ }}$$ (iii) $$\left| \mathcal{S} \right| = 9,{\text{ }}N = 4.$$ If (i) holds then $$\mathcal{S}$$ is spread in a hyperplane.It is shown that possibility (ii) is realized by precisely threeprojectively distinct types of partial spread. Explicit examplesare also given of four projectively distinct types of partialspreads which realize possibility (iii). For one of these types,type X, the four reguli have a common line. It isshown that those partial spreads in PG(4,2) of size 9 which arise, by a simple construction, from a spreadin PG(5,2), are all of type X. Kluwer Academic Publishers, 2000 Subsets of PG( n ,2) nationallicence partial spreads in PG(4,2) nationallicence reguli nationallicence null polarity nationallicence Designs, Codes and Cryptography Kluwer Academic Publishers 21/1-3(2000-10-01), 209-222 0925-1022 21:1-3<209 2000 21 10623 https://doi.org/10.1023/A:1008304115031 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1008304115031 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Shaw Ron Department of Mathematics, University of Hull, HU67RX, Hull, United Kingdom aut NATIONALLICENCE 773 0- Designs, Codes and Cryptography Kluwer Academic Publishers 21/1-3(2000-10-01), 209-222 0925-1022 21:1-3<209 2000 21 10623 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer