On Linear Operator Channels over Finite Fields

Motivated by linear network coding, communication channels perform linear operation over finite fields, namely linear operator channels (LOCs), are studied in this paper. For such a channel, its output vector is a linear transform of its input vector, and the transformation matrix is randomly and independently generated. The transformation matrix is assumed to remain constant for every T input vectors and to be unknown to both the transmitter and the receiver. There are NO constraints on the distribution of the transformation matrix and the field size. Specifically, the optimality of subspace coding over LOCs is investigated. A lower bound on the maximum achievable rate of subspace coding is obtained and it is shown to be tight for some cases. The maximum achievable rate of constant-dimensional subspace coding is characterized and the loss of rate incurred by using constant-dimensional subspace coding is insignificant. The maximum achievable rate of channel training is close to the lower bound on the maximum achievable rate of subspace coding. Two coding approaches based on channel training are proposed and their performances are evaluated. Our first approach makes use of rank-metric codes and its optimality depends on the existence of maximum rank distance codes. Our second approach applies linear coding and it can achieve the maximum achievable rate of channel training. Our code designs require only the knowledge of the expectation of the rank of the transformation matrix. The second scheme can also be realized ratelessly without a priori knowledge of the channel statistics.Comment: 53 pages, 3 figures, submitted to IEEE Transaction on Information Theor

Lossless and near-lossless source coding for multiple access networks

A multiple access source code (MASC) is a source code designed for the following network configuration: a pair of correlated information sequences {X-i}(i=1)(infinity), and {Y-i}(i=1)(infinity) is drawn independent and identically distributed (i.i.d.) according to joint probability mass function (p.m.f.) p(x, y); the encoder for each source operates without knowledge of the other source; the decoder jointly decodes the encoded bit streams from both sources. The work of Slepian and Wolf describes all rates achievable by MASCs of infinite coding dimension (n --> infinity) and asymptotically negligible error probabilities (P-e((n)) --> 0). In this paper, we consider the properties of optimal instantaneous MASCs with finite coding dimension (n 0) performance. The interest in near-lossless codes is inspired by the discontinuity in the limiting rate region at P-e((n)) = 0 and the resulting performance benefits achievable by using near-lossless MASCs as entropy codes within lossy MASCs. Our central results include generalizations of Huffman and arithmetic codes to the MASC framework for arbitrary p(x, y), n, and P-e((n)) and polynomial-time design algorithms that approximate these optimal solutions

Multi-Way Relay Networks: Orthogonal Uplink, Source-Channel Separation and Code Design

We consider a multi-way relay network with an orthogonal uplink and correlated sources, and we characterise reliable communication (in the usual Shannon sense) with a single-letter expression. The characterisation is obtained using a joint source-channel random-coding argument, which is based on a combination of Wyner et al.'s "Cascaded Slepian-Wolf Source Coding" and Tuncel's "Slepian-Wolf Coding over Broadcast Channels". We prove a separation theorem for the special case of two nodes; that is, we show that a modular code architecture with separate source and channel coding functions is (asymptotically) optimal. Finally, we propose a practical coding scheme based on low-density parity-check codes, and we analyse its performance using multi-edge density evolution.Comment: Authors' final version (accepted and to appear in IEEE Transactions on Communications

On the Combinatorial Version of the Slepian-Wolf Problem

We study the following combinatorial version of the Slepian-Wolf coding scheme. Two isolated Senders are given binary strings $X$ and $Y$ respectively; the length of each string is equal to $n$, and the Hamming distance between the strings is at most $\alpha n$. The Senders compress their strings and communicate the results to the Receiver. Then the Receiver must reconstruct both strings $X$ and $Y$. The aim is to minimise the lengths of the transmitted messages. For an asymmetric variant of this problem (where one of the Senders transmits the input string to the Receiver without compression) with deterministic encoding a nontrivial lower bound was found by A.Orlitsky and K.Viswanathany. In our paper we prove a new lower bound for the schemes with syndrome coding, where at least one of the Senders uses linear encoding of the input string. For the combinatorial Slepian-Wolf problem with randomized encoding the theoretical optimum of communication complexity was recently found by the first author, though effective protocols with optimal lengths of messages remained unknown. We close this gap and present a polynomial time randomized protocol that achieves the optimal communication complexity.Comment: 20 pages, 14 figures. Accepted to IEEE Transactions on Information Theory (June 2018

Source-Channel Secrecy with Causal Disclosure

Imperfect secrecy in communication systems is investigated. Instead of using equivocation as a measure of secrecy, the distortion that an eavesdropper incurs in producing an estimate of the source sequence is examined. The communication system consists of a source and a broadcast (wiretap) channel, and lossless reproduction of the source sequence at the legitimate receiver is required. A key aspect of this model is that the eavesdropper's actions are allowed to depend on the past behavior of the system. Achievability results are obtained by studying the performance of source and channel coding operations separately, and then linking them together digitally. Although the problem addressed here has been solved when the secrecy resource is shared secret key, it is found that substituting secret key for a wiretap channel brings new insights and challenges: the notion of weak secrecy provides just as much distortion at the eavesdropper as strong secrecy, and revealing public messages freely is detrimental.Comment: Allerton 2012, 6 pages. Updated version includes acknowledgement

Lossy Compression with Privacy Constraints: Optimality of Polar Codes

A lossy source coding problem with privacy constraint is studied in which two correlated discrete sources $X$ and $Y$ are compressed into a reconstruction $\hat{X}$ with some prescribed distortion $D$. In addition, a privacy constraint is specified as the equivocation between the lossy reconstruction $\hat{X}$ and $Y$. This models the situation where a certain amount of source information from one user is provided as utility (given by the fidelity of its reconstruction) to another user or the public, while some other correlated part of the source information $Y$ must be kept private. In this work, we show that polar codes are able, possibly with the aid of time sharing, to achieve any point in the optimal rate-distortion-equivocation region identified by Yamamoto, thus providing a constructive scheme that obtains the optimal tradeoff between utility and privacy in this framework.Comment: Submitted for publicatio

Perfectly Secure Index Coding

In this paper, we investigate the index coding problem in the presence of an eavesdropper. Messages are to be sent from one transmitter to a number of legitimate receivers who have side information about the messages, and share a set of secret keys with the transmitter. We assume perfect secrecy, meaning that the eavesdropper should not be able to retrieve any information about the message set. We study the minimum key lengths for zero-error and perfectly secure index coding problem. On one hand, this problem is a generalization of the index coding problem (and thus a difficult one). On the other hand, it is a generalization of the Shannon's cipher system. We show that a generalization of Shannon's one-time pad strategy is optimal up to a multiplicative constant, meaning that it obtains the entire boundary of the cone formed by looking at the secure rate region from the origin. Finally, we consider relaxation of the perfect secrecy and zero-error constraints to weak secrecy and asymptotically vanishing probability of error, and provide a secure version of the result, obtained by Langberg and Effros, on the equivalence of zero-error and $\epsilon$-error regions in the conventional index coding problem.Comment: 25 pages, 5 figures, submitted to the IEEE Transactions on Information Theor

Interference Mitigation in Large Random Wireless Networks

A central problem in the operation of large wireless networks is how to deal with interference -- the unwanted signals being sent by transmitters that a receiver is not interested in. This thesis looks at ways of combating such interference. In Chapters 1 and 2, we outline the necessary information and communication theory background, including the concept of capacity. We also include an overview of a new set of schemes for dealing with interference known as interference alignment, paying special attention to a channel-state-based strategy called ergodic interference alignment. In Chapter 3, we consider the operation of large regular and random networks by treating interference as background noise. We consider the local performance of a single node, and the global performance of a very large network. In Chapter 4, we use ergodic interference alignment to derive the asymptotic sum-capacity of large random dense networks. These networks are derived from a physical model of node placement where signal strength decays over the distance between transmitters and receivers. (See also arXiv:1002.0235 and arXiv:0907.5165.) In Chapter 5, we look at methods of reducing the long time delays incurred by ergodic interference alignment. We analyse the tradeoff between reducing delay and lowering the communication rate. (See also arXiv:1004.0208.) In Chapter 6, we outline a problem that is equivalent to the problem of pooled group testing for defective items. We then present some new work that uses information theoretic techniques to attack group testing. We introduce for the first time the concept of the group testing channel, which allows for modelling of a wide range of statistical error models for testing. We derive new results on the number of tests required to accurately detect defective items, including when using sequential `adaptive' tests.Comment: PhD thesis, University of Bristol, 201