2,080 research outputs found
Equiangular Tight Frames from Group Divisible Designs
An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem in many cases. In this paper, we observe that many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known families of ETFs and GDDs, we obtain several new infinite families of ETFs. The real instances of these ETFs correspond to several new infinite families of strongly regular graphs. Our approach was inspired by a seminal paper of Davis and Jedwab which both unified and generalized McFarland and Spence difference sets. We provide combinatorial analogs of their algebraic results, unifying Steiner ETFs with hyperoval ETFs and Tremain ETFs
Equiangular tight frames from group divisible designs
An equiangular tight frame (ETF) is a type of optimal packing of lines in a
real or complex Hilbert space. In the complex case, the existence of an ETF of
a given size remains an open problem in many cases. In this paper, we observe
that many of the known constructions of ETFs are of one of two types. We
further provide a new method for combining a given ETF of one of these two
types with an appropriate group divisible design (GDD) in order to produce a
larger ETF of the same type. By applying this method to known families of ETFs
and GDDs, we obtain several new infinite families of ETFs. The real instances
of these ETFs correspond to several new infinite families of strongly regular
graphs. Our approach was inspired by a seminal paper of Davis and Jedwab which
both unified and generalized McFarland and Spence difference sets. We provide
combinatorial analogs of their algebraic results, unifying Steiner ETFs with
hyperoval ETFs and Tremain ETFs
Problems on Polytopes, Their Groups, and Realizations
The paper gives a collection of open problems on abstract polytopes that were
either presented at the Polytopes Day in Calgary or motivated by discussions at
the preceding Workshop on Convex and Abstract Polytopes at the Banff
International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete
Geometry, to appear
Counting Steiner triple systems with classical parameters and prescribed rank
By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank of
a Steiner triple system on points is at least , and equality
holds only for the classical point-line design in the projective geometry
. It follows from results of Assmus \cite{A} that, given any integer
with , there is a code containing
representatives of all isomorphism classes of STS with 2-rank at most
. Using a mixture of coding theoretic, geometric, design
theoretic and combinatorial arguments, we prove a general formula for the
number of distinct STS with 2-rank at most contained
in this code. This generalizes the only previously known cases, , proved
by Tonchev \cite{T01} in 2001, , proved by V. Zinoviev and D. Zinoviev
\cite{ZZ12} in 2012, and (V. Zinoviev and D. Zinoviev \cite{ZZ13},
\cite{ZZ13a} (2013), D. Zinoviev \cite{Z16} (2016)), while also unifying and
simplifying the proofs. This enumeration result allows us to prove lower and
upper bounds for the number of isomorphism classes of STS with 2-rank
exactly (or at most) . Finally, using our recent systematic
study of the ternary block codes of Steiner triple systems \cite{JT}, we obtain
analogous results for the ternary case, that is, for STS with 3-rank at
most (or exactly) . We note that this work provides the first
two infinite families of 2-designs for which one has non-trivial lower and
upper bounds for the number of non-isomorphic examples with a prescribed
-rank in almost the entire range of possible ranks.Comment: 27 page
- β¦