Some Properties of Strongly Regular Graphs

An approach to the enumeration of feasible parameters for strongly regular graphs is described, based on the pair of structural parameters (a,c) and the positive eigenvalue e. The Krein bound ensures that there are only finitely many possibilities for c, given a and e, and the standard divisibility conditions can be used to reduce the possibilities further. Many sets of feasible parameters appear to be accidents of arithmetic, but in some cases the conditions are satisfied for algebraic reasons. As an example, we discuss an infinite family of feasible parameters for which the corresponding graphs necessarily have a closed neighborhood as a star complement for e.Comment: 15 page

Coded Caching based on Combinatorial Designs

We consider the standard broadcast setup with a single server broadcasting information to a number of clients, each of which contains local storage (called \textit{cache}) of some size, which can store some parts of the available files at the server. The centralized coded caching framework, consists of a caching phase and a delivery phase, both of which are carefully designed in order to use the cache and the channel together optimally. In prior literature, various combinatorial structures have been used to construct coded caching schemes. In this work, we propose a binary matrix model to construct the coded caching scheme. The ones in such a \textit{caching matrix} indicate uncached subfiles at the users. Identity submatrices of the caching matrix represent transmissions in the delivery phase. Using this model, we then propose several novel constructions for coded caching based on the various types of combinatorial designs. While most of the schemes constructed in this work (based on existing designs) have a high cache requirement (uncached fraction being $\Theta(\frac{1}{\sqrt{K}})$ or $\Theta(\frac{1}{K})$, $K$ being the number of users), they provide a rate that is either constant or decreasing ($O(\frac{1}{K})$) with increasing $K$, and moreover require competitively small levels of subpacketization (being $O(K^i), 1\leq i\leq 3$), which is an extremely important parameter in practical applications of coded caching. We mark this work as another attempt to exploit the well-developed theory of combinatorial designs for the problem of constructing caching schemes, utilizing the binary caching model we develop.Comment: 10 pages, Appeared in Proceedings of IEEE ISIT 201

Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms

We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. Further, we discuss trigeodetic graphs obtained from the symmetric $(45,12,3)$ designs. We prove that $k$-geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs obtained from symmetric $(45,12,3)$ designs

Resolving sets for Johnson and Kneser graphs

A set of vertices $S$ in a graph $G$ is a {\em resolving set} for $G$ if, for any two vertices $u,v$, there exists $x\in S$ such that the distances $d(u,x) \neq d(v,x)$. In this paper, we consider the Johnson graphs $J(n,k)$ and Kneser graphs $K(n,k)$, and obtain various constructions of resolving sets for these graphs. As well as general constructions, we show that various interesting combinatorial objects can be used to obtain resolving sets in these graphs, including (for Johnson graphs) projective planes and symmetric designs, as well as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems and toroidal grids.Comment: 23 pages, 2 figures, 1 tabl

Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms

We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. Further, we discuss trigeodetic graphs obtained from the symmetric $(45,12,3)$ designs. We prove that $k$-geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs obtained from symmetric $(45,12,3)$ designs