The problem of reconstructing a source sequence with the presence of decoder side-information that is mis-synchronized to the source due to deletions is studied in a distributed source coding framework. Motivated by practical applications, the deletion process is assumed to be bursty and is modeled by a Markov chain. The minimum rate needed to reconstruct the source sequence with high probability is characterized in terms of an information theoretic expression, which is interpreted as the amount of information of the deleted content and the locations of deletions, subtracting "nature's secret", that is, the uncertainty of the locations given the source and side-information. For small bursty deletion probability, the asymptotic expansion of the minimum rate is computed.
Suppose that we have two similar files stored on different computers. We need to send the file from the first computer to the second one trying to minimize the number of bits transmitted. This article presents a survey of results known for this communication complexity problem in the case when files are "similar" in the sense of Hamming distance. We mainly systematize earlier results obtained by various authors in 1990s and 2000s and discuss its connection with coding theory, hashing algorithms and other domains of computer science. In particular cases we propose some improvements of previous …
Differential distributed space-time coding (D-DSTC) has been considered to improve both diversity and data-rate in cooperative communications in the absence of channel information. However, conventionally, it is assumed that relays are perfectly synchronized in the symbol level. In practice, this assumption is easily violated due to the distributed nature of the relay networks. This paper proposes a new differential encoding and decoding process for D-DSTC systems with two relays. The proposed method is robust against synchronization errors and does not require any channel information at the d…
Network Coding is a packet encoding technique which has recently been shown to improve network performance (by reducing delays and increasing throughput) in broadcast and multicast communications. The cost for such an improvement comes in the form of increased decoding complexity (and thus delay) at the receivers end. Before delivering the file to higher layers, the receiver should first decode those packets. In our work we consider the broadcast transmission of a large file to N wireless users. The file is segmented into a number of blocks (each containing K packets - the Coding Window Size).…
We study the broadcast transmission of a single file to an arbitrary number of receivers using Random Linear Network Coding (RLNC) in a network with unreliable channels. Due to the increased computational complexity of the decoding process (especially for large files) we apply chunked RLNC (i.e. RLNC is applied within non-overlapping subsets of the file). In our work we show the optimality of the Least Received (LR) batch scheduling policy (which was introduced in our prior work) with regards to the expected file transfer completion time. Furthermore, we refine some of our earlier results, nam…
Differential distributed space-time coding (D-DSTC) is a cooperative transmission technique that can improve diversity in wireless relay networks in the absence of channel information. Conventionally, it is assumed that channels are flat-fading and relays are perfectly synchronized at the symbol level. However, due to the delay spread in broadband systems and the distributed nature of relay networks, these assumptions may be violated. Hence, inter-symbol interference (ISI) may appear. This paper proposes a new differential encoding and decoding process for D-DSTC systems with multiple relays o…
We consider a system consisting of a large network of $n$ users and a library of files, wherein inter-user communication is established based upon gossip mechanisms. Each file is initially present at exactly one node, which is designated as the file \emph{source}. The source gets updated with newer versions of the file according to an arbitrary distribution in real time, and the other users in the network wish to acquire the latest possible version of the file. We present a class of gossip protocols that achieve $O(1)$ age at a typical node in a single-file system and $O(n)$ age at a typical n…
Content distribution over networks is often achieved by using mirror sites that hold copies of files or portions thereof to avoid congestion and delay issues arising from excessive demands to a single location. Accordingly, there are distributed storage solutions that divide the file into pieces and place copies of the pieces (replication) or coded versions of the pieces (coding) at multiple source nodes. We consider a network which uses network coding for multicasting the file. There is a set of source nodes that contains either subsets or coded versions of the pieces of the file. The cost of…
We present many new results related to reliable (interactive) communication over insertion-deletion channels. Synchronization errors, such as insertions and deletions, strictly generalize the usual symbol corruption errors and are much harder to protect against. We show how to hide the complications of synchronization errors in many applications by introducing very general channel simulations which efficiently transform an insertion-deletion channel into a regular symbol corruption channel with an error rate larger by a constant factor and a slightly smaller alphabet. We generalize synchroniza…
To tackle the expected enormous increase in mobile video traffic in cellular networks, an architecture involving a base station along with caching femto stations (referred to as helpers), storing popular files near users, has been proposed [1]. The primary benefit of caching is the enormous increase in downloading rate when a popular file is available at helpers near a user requesting that file. In this work, we explore a secondary benefit of caching in this architecture through the lens of index coding. We assume a system with n users and constant number of caching helpers. Only helpers store…
Folding a sequence $S$ into a multidimensional box is a method that is used to construct multidimensional codes. The well known operation of folding is generalized in a way that the sequence $S$ can be folded into various shapes. The new definition of folding is based on lattice tiling and a direction in the $D$-dimensional grid. There are potentially $\frac{3^D-1}{2}$ different folding operations. Necessary and sufficient conditions that a lattice combined with a direction define a folding are given. The immediate and most impressive application is some new lower bounds on the number of dots …
Sidon sequences and their generalizations have found during the years and especially recently various applications in coding theory. One of the most important applications of these sequences is in the connection of synchronization patterns. A few constructions of two-dimensional synchronization patterns are based on these sequences. In this paper we present sufficient conditions that a two-dimensional synchronization pattern can be transformed into a Sidon sequence. We also present a new construction for Sidon sequences over an alphabet of size q(q-1), where q is a power of a prime.
This paper considers the problem of transferring a file from one source node to multiple receivers in a peer-to-peer (P2P) network. The objective is to minimize the weighted sum download time (WSDT) for the one-to-many file transfer. Previous work has shown that, given an order at which the receivers finish downloading, the minimum WSD can be solved in polynomial time by convex optimization, and can be achieved by linear network coding, assuming that node uplinks are the only bottleneck in the network. This paper, however, considers heterogeneous peers with both uplink and downlink bandwidth c…
This paper presents a novel approach to network coding for distribution of large files. Instead of the usual approach of splitting packets into disjoint classes (also known as generations) we propose the use of overlapping classes. The overlapping allows the decoder to alternate between Gaussian elimination and back substitution, simultaneously boosting the performance and reducing the decoding complexity. Our approach can be seen as a combination of fountain coding and network coding. Simulation results are presented that demonstrate the promise of our approach.
Folding a sequence $S$ into a multidimensional box is a well-known method which is used as a multidimensional coding technique. The operation of folding is generalized in a way that the sequence $S$ can be folded into various shapes and not just a box. The new definition of folding is based on a lattice tiling for the given shape $\cS$ and a direction in the $D$-dimensional integer grid. Necessary and sufficient conditions that a lattice tiling for $\cS$ combined with a direction define a folding of a sequence into $\cS$ are derived. The immediate and most impressive application is some new lo…
We consider the problem of constructing codes that can correct $δ$ deletions occurring in an arbitrary binary string of length $n$ bits. Varshamov-Tenengolts (VT) codes can correct all possible single deletions $(δ=1)$ with an asymptotically optimal redundancy. Finding similar codes for $δ\geq 2$ deletions is an open problem. We propose a new family of codes, that we call Guess & Check (GC) codes, that can correct, with high probability, a constant number of deletions $δ$ occurring at uniformly random positions within an arbitrary string. The GC codes are based on MDS codes and have an asympto…
In this paper, we consider a synchronization problem between nodes $A$ and $B$ that are connected through a two--way communication channel. {Node $A$} contains a binary file $X$ of length $n$ and {node $B$} contains a binary file $Y$ that is generated by randomly deleting bits from $X$, by a small deletion rate $β$. The location of deleted bits is not known to either node $A$ or node $B$. We offer a deterministic synchronization scheme between nodes $A$ and $B$ that needs a total of $O(nβ\log \frac{1}β)$ transmitted bits and reconstructs $X$ at node $B$ with probability of error that is expone…
The capacity of caching networks has received considerable attention in the past few years. A particularly studied setting is the case of a single server (e.g., a base station) and multiple users, each of which caches segments of files in a finite library. Each user requests one (whole) file in the library and the server sends a common coded multicast message to satisfy all users at once. The problem consists of finding the smallest possible codeword length to satisfy such requests. In this paper we consider the generalization to the case where each user places $L \geq 1$ requests. The obvious…
A cache-aided $K$-user Gaussian broadcast channel (BC) is studied. The transmitter has a library of $N$ files, from which each user requests one. The users are equipped with caches of different sizes, which are filled without the knowledge of the user requests in a centralized manner. Differently from the literature, it is assumed that each file can be delivered to different users at different rates, which may correspond to different quality representations of the underlying content, e.g., scalable coded video segments. Accordingly, instead of a single achievable rate, the system performance i…
We consider the problem of reconciling similar, but remote, strings with minimum communication complexity. This "string reconciliation" problem is a fundamental building block for a variety of networking applications, including those that maintain large-scale distributed networks and perform remote file synchronization. We present the novel Recursive Content-Dependent Shingling (RCDS) protocol that is computationally practical for large strings and scales linearly with the edit distance between the remote strings. We provide comparisons to the performance of Rsync, one of the most popular file…
We consider the problem of designing decentralized schemes for coded caching. In this problem there are $K$ users each caching $M$ files out of a library of $N$ total files. The question is to minimize $R$, the number of broadcast transmissions to satisfy all the user demands. Decentralized schemes allow the creation of each cache independently, allowing users to join or leave without dependencies. Previous work showed that to achieve a coding gain $g$, i.e. $R \leq K (1-M/N)/g$ transmissions, each file has to be divided into number of subpackets that is exponential in $g$. In this work we pro…
We consider the problem of designing codes for distributed storage that protect user data against eavesdroppers that can gain access to network links as well as individual nodes. Our goal is to achieve weak security (also known as block security) that requires that the eavesdroppers would not be able to decode individual files or combinations of a small number of files. The standard approach for achieving block security is to use a joint design scheme that consists of (inner) storage code and the (outer) coset code. However, jointly designing the codes requires that the user, who pre-processes…
This paper studies the fundamental limits of content delivery in a cache-aided broadcast network for correlated content generated by a discrete memoryless source with arbitrary joint distribution. Each receiver is equipped with a cache of equal capacity, and the requested files are delivered over a shared error-free broadcast link. A class of achievable correlation-aware schemes based on a two-step source coding approach is proposed. Library files are first compressed, and then cached and delivered using a combination of correlation-unaware multiple-request cache-aided coded multicast schemes.…
A client/encoder edits a file, as modeled by an insertion-deletion (InDel) process. An old copy of the file is stored remotely at a data-centre/decoder, and is also available to the client. We consider the problem of throughput- and computationally-efficient communication from the client to the data-centre, to enable the server to update its copy to the newly edited file. We study two models for the source files/edit patterns: the random pre-edit sequence left-to-right random InDel (RPES-LtRRID) process, and the arbitrary pre-edit sequence arbitrary InDel (APES-AID) process. In both models, we…
We consider the canonical {\em shared link network} formed by a source node, hosting a library of $m$ information messages (files), connected via a noiseless common link to $n$ destination nodes (users), each with a cache of size M files. Users request files at random and independently, according to a given a-priori demand distribution $\qv$. A coding scheme for this network consists of a caching placement (i.e., a mapping of the library files into the user caches) and delivery scheme (i.e., a mapping for the library files and user demands into a common multicast codeword) such that, after the…
Locating the boundaries of consecutive blocks of quantum information is a fundamental building block for advanced quantum computation and quantum communication systems. We develop a coding theoretic method for properly locating boundaries of quantum information without relying on external synchronization when block synchronization is lost. The method also protects qubits from decoherence in a manner similar to conventional quantum error-correcting codes, seamlessly achieving synchronization recovery and error correction. A family of quantum codes that are simultaneously synchronizable and erro…
We consider the problem of constructing codes that can correct $δ$ deletions occurring in an arbitrary binary string of length $n$ bits. Varshamov-Tenengolts (VT) codes, dating back to 1965, are zero-error single deletion $(δ=1)$ correcting codes, and have an asymptotically optimal redundancy. Finding similar codes for $δ\geq 2$ deletions remains an open problem. In this work, we relax the standard zero-error (i.e., worst-case) decoding requirement by assuming that the positions of the $δ$ deletions (or insertions) are independent of the codeword. Our contribution is a new family of explicit c…
Cache-aided coded multicast leverages side information at wireless edge caches to efficiently serve multiple unicast demands via common multicast transmissions, leading to load reductions that are proportional to the aggregate cache size. However, the increasingly dynamic, unpredictable, and personalized nature of the content that users consume challenges the efficiency of existing caching-based solutions in which only exact content reuse is explored. This paper generalizes the cache-aided coded multicast problem to specifically account for the correlation among content files, such as, for exa…
In this paper we close the gap between end-to-end diversity coding and intra-session network coding for unicast connections resilient against single link failures. In particular, we show that coding operations are sufficient to perform at the source and receiver if the user data can be split into at most two parts over the filed GF(2). Our proof is purely combinatorial and based on standard graph and network flow techniques. It is a linear time construction that defines the route of subflows A, B and A+B between the source and destination nodes. The proposed resilient flow decomposition method…
We consider the problem of distributing a file in a network of storage nodes whose storage budget is limited but at least equals to the size file. We first generate $T$ encoded symbols (from the file) which are then distributed among the nodes. We investigate the optimal allocation of $T$ encoded packets to the storage nodes such that the probability of reconstructing the file by using any $r$ out of $n$ nodes is maximized. Since the optimal allocation of encoded packets is difficult to find in general, we find another objective function which well approximates the original problem and yet is …
The correlation among the content distributed across a cache-aided broadcast network can be exploited to reduce the delivery load on the shared wireless link. This paper considers a two-user three-file network with correlated content, and studies its fundamental limits for the worst-case demand. A class of achievable schemes based on a two-step source coding approach is proposed. Library files are first compressed using Gray-Wyner source coding, and then cached and delivered using a combination of correlation-unaware cache-aided coded multicast schemes. The second step is interesting in its ow…
In this paper, we propose a polar coding based scheme for set reconciliation between two network nodes. The system is modeled as a well-known Slepian-Wolf setting induced by a fixed number of deletions. The set reconciliation process is divided into two phases: 1) a deletion polar code is employed to help one node to identify the possible deletion indices, which may be larger than the number of genuine deletions; 2) a lossless compression polar code is then designed to feedback those indices with minimum overhead. Our scheme can be viewed as a generalization of polar codes to some emerging net…
Consider two remote nodes (encoder and decoder), each with a binary sequence. The encoder's sequence $X$ differs from the decoder's sequence $Y$ by a small number of edits (deletions and insertions). The goal is to construct a message $M$, to be sent via a one-way error free link, such that the decoder can reconstruct $X$ using $M$ and $Y$. In this paper, we devise a coding scheme for this one-way synchronization model. The scheme is based on multiple layers of Varshamov-Tenengolts (VT) codes combined with off-the-shelf linear error-correcting codes, and uses a list decoder. We bound the expec…
In this paper, we present two new ways of quantum synchronization coding based on the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction and the product construction respectively, and greatly enrich the varieties of available quantum synchronizable codes. The circumstances where the maximum synchronization error tolerance can be reached are explained for both constructions. Furthermore, repeated-root cyclic codes derived from the $(\bm{u}+\bm{v}|\bm{u}-\bm{v})$ construction are shown to be able to provide better Pauli error-correcting capability than BCH codes.
