Difference systems of sets (DSSs) are combinatorial structures introduced by Levenshtein in connection with code synchronization. In this paper, some recursive constructions of DSSs obtained from ...finite projective geometry are presented. As a consequence, new infinite families of optimal DSSs are obtained.
A fundamental question for simplicial complexes is to find the lowest dimensional Euclidean space in which they can be embedded. We investigate this question for order complexes of posets. We show ...that order complexes of thick geometric lattices as well as several classes of finite buildings, all of which are order complexes, are hard to embed. That means that such
d
-dimensional complexes require
(
2
d
+
1
)
-dimensional Euclidean space for an embedding. (This dimension is always sufficient for any
d
-complex.) We develop a method to show non-embeddability for general order complexes of posets.
A spread in PG(n, q) is a set of lines such that each point is in exactly one line. A parallelism is a partition of the set of lines of PG(n, q) to spreads. The construction of spreads and ...parallelisms is motivated by their various relations and applications. Most of the presently known explicit constructions of parallelisms are for n = 3. PG(5, 2) is the smallest projective space with n = 5. All point-transitive and all cyclic parallelisms of PG(5, 2) are known. In the present work we establish that up to conjugacy there is only one cyclic subgroup (of the automorphism group of the projective space) of order 21 which can be the automorphism group of a parallelism. We construct all parallelisms invariant under this subgroup that have the greatest possible number of spreads fixed under the assumed subgroup. We compute the automorphism group orders and invariants of the spreads of all the 2138 parallelisms of PG(5, 2) which we obtain.
There are 372 parallelisms of PG(3,5) which have been explicitly constructed and studied before the present paper. They contain all but one of the 21 nonisomorphic spreads of this projective space. ...It was not known by now if that particular spread can be included in a parallelism. We give a positive answer to this question by constructing parallelisms invariant under a cyclic group of order 8 some of which contain the spread. Useful invariants are calculated for all the obtained 899 new parallelisms of PG(3,5).
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