Locally Encodable and Decodable Codes for Distributed Storage Systems

We consider the locality of encoding and decoding operations in distributed storage systems (DSS), and propose a new class of codes, called locally encodable and decodable codes (LEDC), that provides a higher degree of operational locality compared to currently known codes. For a given locality structure, we derive an upper bound on the global distance and demonstrate the existence of an optimal LEDC for sufficiently large field size. In addition, we also construct two families of optimal LEDC for fields with size linear in code length.Comment: 7 page

On the Capacity Region for Secure Index Coding

We study the index coding problem in the presence of an eavesdropper, where the aim is to communicate without allowing the eavesdropper to learn any single message aside from the messages it may already know as side information. We establish an outer bound on the underlying secure capacity region of the index coding problem, which includes polymatroidal and security constraints, as well as the set of additional decoding constraints for legitimate receivers. We then propose a secure variant of the composite coding scheme, which yields an inner bound on the secure capacity region of the index coding problem. For the achievability of secure composite coding, a secret key with vanishingly small rate may be needed to ensure that each legitimate receiver who wants the same message as the eavesdropper, knows at least two more messages than the eavesdropper. For all securely feasible index coding problems with four or fewer messages, our numerical results establish the secure index coding capacity region

Optimal Index Codes via a Duality between Index Coding and Network Coding

In Index Coding, the goal is to use a broadcast channel as efficiently as possible to communicate information from a source to multiple receivers which can possess some of the information symbols at the source as side-information. In this work, we present a duality relationship between index coding (IC) and multiple-unicast network coding (NC). It is known that the IC problem can be represented using a side-information graph $G$ (with number of vertices $n$ equal to the number of source symbols). The size of the maximum acyclic induced subgraph, denoted by $MAIS$ is a lower bound on the \textit{broadcast rate}. For IC problems with $MAIS=n-1$ and $MAIS=n-2$, prior work has shown that binary (over ${\mathbb F}_2$) linear index codes achieve the $MAIS$ lower bound for the broadcast rate and thus are optimal. In this work, we use the the duality relationship between NC and IC to show that for a class of IC problems with $MAIS=n-3$, binary linear index codes achieve the $MAIS$ lower bound on the broadcast rate. In contrast, it is known that there exists IC problems with $MAIS=n-3$ and optimal broadcast rate strictly greater than $MAIS$

TDMA is Optimal for All-unicast DoF Region of TIM if and only if Topology is Chordal Bipartite

The main result of this work is that an orthogonal access scheme such as TDMA achieves the all-unicast degrees of freedom (DoF) region of the topological interference management (TIM) problem if and only if the network topology graph is chordal bipartite, i.e., every cycle that can contain a chord, does contain a chord. The all-unicast DoF region includes the DoF region for any arbitrary choice of a unicast message set, so e.g., the results of Maleki and Jafar on the optimality of orthogonal access for the sum-DoF of one-dimensional convex networks are recovered as a special case. The result is also established for the corresponding topological representation of the index coding problem

On Approximating the Sum-Rate for Multiple-Unicasts

We study upper bounds on the sum-rate of multiple-unicasts. We approximate the Generalized Network Sharing Bound (GNS cut) of the multiple-unicasts network coding problem with $k$ independent sources. Our approximation algorithm runs in polynomial time and yields an upper bound on the joint source entropy rate, which is within an $O(\log^2 k)$ factor from the GNS cut. It further yields a vector-linear network code that achieves joint source entropy rate within an $O(\log^2 k)$ factor from the GNS cut, but \emph{not} with independent sources: the code induces a correlation pattern among the sources. Our second contribution is establishing a separation result for vector-linear network codes: for any given field $\mathbb{F}$ there exist networks for which the optimum sum-rate supported by vector-linear codes over $\mathbb{F}$ for independent sources can be multiplicatively separated by a factor of $k^{1-\delta}$, for any constant ${\delta>0}$, from the optimum joint entropy rate supported by a code that allows correlation between sources. Finally, we establish a similar separation result for the asymmetric optimum vector-linear sum-rates achieved over two distinct fields $\mathbb{F}_{p}$ and $\mathbb{F}_{q}$ for independent sources, revealing that the choice of field can heavily impact the performance of a linear network code.Comment: 10 pages; Shorter version appeared at ISIT (International Symposium on Information Theory) 2015; some typos correcte

Locality in Index Coding for Large Min-Rank

An index code is said to be locally decodable if each receiver can decode its demand using its side information and by querying only a subset of the transmitted codeword symbols instead of observing the entire codeword. Local decodability can be a beneficial feature in some communication scenarios, such as when the receivers can afford to listen to only a part of the transmissions because of limited availability of power. The locality of an index code is the ratio of the maximum number of codeword symbols queried by a receiver to the message length. In this paper we analyze the optimum locality of linear codes for the family of index coding problems whose min-rank is one less than the number of receivers in the network. We first derive the optimal trade-off between the index coding rate and locality with vector linear coding when the side information graph is a directed cycle. We then provide the optimal trade-off achieved by scalar linear coding for a larger family of problems, viz., problems where the min-rank is only one less than the number of receivers. While the arguments used for achievability are based on known coding techniques, the converse arguments rely on new results on the structure of locally decodable index codes.Comment: Keywords: index codes, locality, min-rank, directed cycle, side information grap