Configurations of planes in PG(5,2)
Discrete Mathematics
Designs, Codes and Cryptography
The cubic Segre variety in PG(5, 2)
Designs, Codes and Cryptography
| Hi-index | 0.00 |
Put \theta_{n}=\#\{pointsin \mbox{PG}(n,2)\} and \phi_{n} =\#\{linesin \mbox{PG}(n,2)\}. Let \psi be anypoint-subset of \mbox{PG}(n,2). It is shown thatthe sum of L=\#\{internal lines of \psi\}and L^{\prime}=\#\{external lines of \psi\}is the same for all \psi having the same cardinality:[6pt] Theorem AIf k is defined by k=|\psi|-\theta_{n-1},then L+L^prime=\phi_n-1+k(k-1)/2.(The generalization of this to subsets of \mbox{PG}(n,3)is also obtained.) Let {\cal S} be a partial spreadof lines in \mbox{PG}(4,2) and let Ndenote the number of reguli contained in {\cal S}.Use of Theorem A gives rise to a simple proof of:[6pt] TheoremBIf {\cal S} is maximal then one of the followingholds: (i) |{\cal S}|=\hspace{3pt}5 ,\;N=\hspace{3pt}10;\quad(ii) |{\cal S}|=\hspace{3pt}7,\;N=\hspace{3pt}4 ;\quad(iii) |{\calS}|=\hspace{3pt}9 ,\;N=\hspace{3pt}4 .If (i) holds then {\cal 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 \mbox{PG}(4,2)of size 9 which arise, by a simple construction, from a spreadin \mbox{PG}(5,2), are all of type X.