Terence Chan

Dualities Between Entropy Functions and Network Codes

In communications networks, the capacity region of multisource network coding is given in terms of the set of entropy functions Gamma*. More broadly, determination of Gamma* would have an impact on converse theorems for multi-terminal problems in information theory. This paper provides several new dualities between entropy functions and network codes. Given a function g ges 0 defined on all subsets of N random variables, we provide a construction for a network multicast problem which is ldquosolvablerdquo if and only if g is the entropy function of a set of quasi-uniform random variables. The underlying network topology is fixed and the multicast problem depends on g only through link capacities and source rates. A corresponding duality is developed for linear network codes, where the constructed multicast problem is linearly solvable if and only if g is linear group characterizable. Relaxing the requirement that the domain of g be subsets of random variables, we obtain a similar duality between polymatroids and the linear programming bound. These duality results provide an alternative proof of the insufficiency of linear (and abelian) network codes, and demonstrate the utility of non-Shannon inequalities to tighten outer bounds on network coding capacity regions.

Theory of Secure Network Coding

In this tutorial paper, we focus on the basic theory of linear secure network coding. Our goal is to present fundamental results and provide preliminary knowledge for anyone interested in the area. We first present a model for secure network coding and then a necessary and sufficient condition for a linear network code to be secure. Optimal methods to construct linear secure network codes are also provided. For further investigation of the secure properties of linear network codes, we illuminate different secure criteria and requirements, with a few alternative models.

Private information retrieval for coded storage

Private information retrieval scheme for coded data storage is considered in this paper. We focus on the case where the size of each data record is large and hence only the download cost (but not the upload cost for transmitting retrieval queries) is of interest. We prove that the tradeoff between storage cost and retrieval/download cost depends on the number of data records in the system. We propose a class of linear storage codes and retrieval schemes, and derive conditions under which our schemes are error-free and private. Tradeoffs between the storage cost and retrieval costs are also obtained.

Rate Distortion With Side-Information at Many Decoders

We present an achievable rate region for the multistage successive-refinement problem with side-information. We also present an upper bound for the rate-distortion function for lossy source coding with side-information at many decoders. Characterising this rate-distortion function is a long-standing open problem, and it is widely believed that the tightest upper bound is provided by Theorem 2 of Heegard and Bergers paper “Rate distortion when side information may be absent” (IEEE Trans. Inf. Theory, 1985). We give a counterexample to Heegard and Bergers result.

Network coding capacity: A functional dependence bound

Explicit characterization and computation of the multi-source network coding capacity region (or even bounds) is long standing open problem. In fact, finding the capacity region requires determination of the set of all entropic vectors Γ*, which is known to be an extremely hard problem. On the other hand, calculating the explicitly known linear programming bound is very hard in practice due to an exponential growth in complexity as a function of network size. We give a new, easily computable outer bound, based on characterization of all functional dependencies in networks. We also show that the proposed bound is tighter than some known bounds.

Mission impossible: Computing the network coding capacity region

One of the main theoretical motivations for the emerging area of network coding is the achievability of the max-flow/min-cut rate for single source multicast. This can exceed the rate achievable with routing alone, and is achievable with linear network codes. The multi-source problem is more complicated. Computation of its capacity region is equivalent to determination of the set of all entropy functions Gamma*, which is non-polyhedral. The aim of this paper is to demonstrate that this difficulty can arise even in single source problems. In particular, for single source networks with hierarchical sink requirements, and for single source networks with secrecy constraints. In both cases, we exhibit networks whose capacity regions involve Gamma*. As in the multi-source case, linear codes are insufficient.

Capacity bounds for secure network coding

We consider the problem of how to securely communicate over networks subject to presence of eavesdroppers. We obtain inner and outer bounds for the set of rate-capacity tuples at which data can be robustly and securely transmitted across an acyclic network under the assumption of error-free links. These bounds generalize the inner and outer bounds obtained for network coding given by Yeung.

Group characterizable entropy functions

This paper studies properties of entropy functions that are induced by groups and subgroups. We showed that many information theoretic properties of those group induced entropy functions also have corresponding group theoretic interpretations. Then we propose an extension method to find outer bound for these group induced entropy functions.

Irregular Fractional Repetition Code Optimization for Heterogeneous Cloud Storage

This paper presents a flexible irregular model for heterogeneous cloud storage systems and investigates how the cost of repairing failed nodes can be minimized. The fractional repetition code, originally designed for minimizing repair bandwidth for homogeneous storage systems, is generalized to the irregular fractional repetition code, which is adaptable to heterogeneous environments. The code structure and the associated storage allocation can be obtained by solving an integer linear programming problem. For moderate sized networks, a heuristic algorithm is proposed and shown to be near-optimal by computer simulations.

On capacity regions of non-multicast networks

We study the network coding capacity of multi-source, multi-sink networks with colocated sources, but where each sink may demand a different subset of the sources. We show that in this scenario, the set of admissible (zero probability of decoding errors) and achievable (vanishing probability of decoding errors) rate capacity tuples are the same. We also simplify the capacity region by showing that the outer bound obtained in “A First Course in Information Theory” (Yeung, 2002) is in fact tight. We conjecture that this bound remains tight, even when the sources are not colocated.