Independent Sets in Regular Hypergraphs and Multidimensional Runlength-Limited Constraints

Abstract

Let G be a t-uniform s-regular linear hypergraph with r vertices. It is shown that the number of independent sets $\IS(\hgraph)$ in $\hgraph$ satisfies \[ \log_2 \IS(\hgraph) \le \frac{r}{t} \left( 1 + O \biggl( \frac{\log^2(ts)}{s} \biggr) \right) . \] This leads to an improvement of a previous bound by Alon obtained for t = 2 (i.e., for regular ordinary graphs). It is also shown that for the Hamming graph $\Hamming(n,q)$ (with vertices consisting of all n-tuples over an alphabet of size q and edges connecting pairs of vertices with Hamming distance $1$), \[ \frac{\log_2 \IS(\Hamming(n,q))}{q^n} = \frac{1}{q} + O \biggl(\frac{\log^2 (q n)}{q n} \biggr). \] The latter result is then applied to show that the Shannon capacity of the n-dimensional $(d,\infty)$-runlength-limited (RLL) constraint converges to 1/(d+1) as n goes to infinity.