Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Binary Codes

Given positive integers $n$ and $d$, let $A_2(n,d)$ denote the maximum size of a binary code of length $n$ and minimum distance $d$. The well-known Gilbert-Varshamov bound asserts that $A_2(n,d) \geq 2^n/V(n,d-1)$, where $V(n,d) = \sum_{i=0}^{d} {n \choose i}$ is the volume of a Hamming sphere of radius $d$. We show that, in fact, there exists a positive constant $c$ such that $$A_2(n,d) \geq c \frac{2^n}{V(n,d-1)} \log_2 V(n,d-1)$$ whenever $d/n \le 0.499$. The result follows by recasting the Gilbert- Varshamov bound into a graph-theoretic framework and using the fact that the corresponding graph is locally sparse. Generalizations and extensions of this result are briefly discussed.Comment: 10 pages, 3 figures; to appear in the IEEE Transactions on Information Theory, submitted August 12, 2003, revised March 28, 200

Multipartite purification protocols: upper and optimal bounds

A method for producing an upper bound for all multipartite purification protocols is devised, based on knowing the optimal protocol for purifying bipartite states. When applied to a range of noise models, both local and correlated, the optimality of certain protocols can be demonstrated for a variety of graph and valence bond states.Comment: 15 pages, 16 figures. v3: published versio

Learning loopy graphical models with latent variables: Efficient methods and guarantees

The problem of structure estimation in graphical models with latent variables is considered. We characterize conditions for tractable graph estimation and develop efficient methods with provable guarantees. We consider models where the underlying Markov graph is locally tree-like, and the model is in the regime of correlation decay. For the special case of the Ising model, the number of samples $n$ required for structural consistency of our method scales as $n=\Omega(\theta_{\min}^{-\delta\eta(\eta+1)-2}\log p)$, where p is the number of variables, $\theta_{\min}$ is the minimum edge potential, $\delta$ is the depth (i.e., distance from a hidden node to the nearest observed nodes), and $\eta$ is a parameter which depends on the bounds on node and edge potentials in the Ising model. Necessary conditions for structural consistency under any algorithm are derived and our method nearly matches the lower bound on sample requirements. Further, the proposed method is practical to implement and provides flexibility to control the number of latent variables and the cycle lengths in the output graph.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1070 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the ultimate normalized chromatic difference sequence of a graph

AbstractFor graphs G and H, the Cartesian product G × H is defined as follows: the vertex set is V(G) × V(H), and two vertices (g,h) and (g′,h′) are adjacent in G × H if either g = g′ and hh′ ϵ E(H) or h = h′ and gg′ ϵ E(G). Let Gk denote the Cartesian product of k copies of G. The chromatic difference sequence cds(G) is defined by cds(G) = (a1, a2 − a1,…, at − at − 1,…) where at denotes the maximum number of vertices of t-colorable subgraph of G. The normalized chromatic difference sequence ncds(G) is defined by ncds(G) = cds(G)/V(G). This paper studies the ultimate normalized chromatic difference sequence of a graph NCDS(G) which is equal to the limit of ncds(Gk) as k goes to infinity. We study NCDS(G) under the context of other graph theoretical properties: star chromatic number, hom-regularity, and graph homomorphism. We have provided new upper and lower bounds for NCDS(G). We have also proved, among others, that if there is a homomorphism from a graph G to a graph H, then NCDS(G) dominates NCDS(H)