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 $2^n-1$ points is at least $2^n -1 -n$, and equality holds only for the classical point-line design in the projective geometry $PG(n-1,2)$. It follows from results of Assmus \cite{A} that, given any integer $t$ with $1 \leq t \leq n-1$, there is a code $C_{n,t}$ containing representatives of all isomorphism classes of STS$(2^n-1)$ with 2-rank at most $2^n -1 -n + t$. Using a mixture of coding theoretic, geometric, design theoretic and combinatorial arguments, we prove a general formula for the number of distinct STS$(2^n-1)$ with 2-rank at most $2^n -1 -n + t$ contained in this code. This generalizes the only previously known cases, $t=1$, proved by Tonchev \cite{T01} in 2001, $t=2$, proved by V. Zinoviev and D. Zinoviev \cite{ZZ12} in 2012, and $t=3$ (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$(2^n-1)$ with 2-rank exactly (or at most) $2^n -1 -n + t$. 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$(3^n)$ with 3-rank at most (or exactly) $3^n -1 -n + t$. 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 $p$-rank in almost the entire range of possible ranks.Comment: 27 page

The Intersection problem for 2-(v; 5; 1) directed block designs

The intersection problem for a pair of 2-(v, 3, 1) directed designs and 2-(v, 4, 1) directed designs is solved by Fu in 1983 and by Mahmoodian and Soltankhah in 1996, respectively. In this paper we determine the intersection problem for 2-(v, 5, 1) directed designs.Comment: 17 pages. To appear in Discrete Mat

Dispersive Elastodynamics of 1D Banded Materials and Structures: Design

Within periodic materials and structures, wave scattering and dispersion occur across constituent material interfaces leading to a banded frequency response. In an earlier paper, the elastodynamics of one-dimensional periodic materials and finite structures comprising these materials were examined with an emphasis on their frequency-dependent characteristics. In this work, a novel design paradigm is presented whereby periodic unit cells are designed for desired frequency band properties, and with appropriate scaling, these cells are used as building blocks for forming fully periodic or partially periodic structures with related dynamical characteristics. Through this multiscale dispersive design methodology, which is hierarchical and integrated, structures can be devised for effective vibration or shock isolation without needing to employ dissipative damping mechanisms. The speed of energy propagation in a designed structure can also be dictated through synthesis of the unit cells. Case studies are presented to demonstrate the effectiveness of the methodology for several applications. Results are given from sensitivity analyses that indicate a high level of robustness to geometric variation.Comment: 33 text pages, 27 figure