arXiv subjects

Andrew Thangaraj

Publications and source records attributed to Andrew Thangaraj.

At least 37 records · Page 2Linked to original sources

arXiv2015

Construction of Near-Capacity Protograph LDPC Code Sequences with Block-Error Thresholds

Density evolution for protograph Low-Density Parity-Check (LDPC) codes is considered, and it is shown that the message-error rate falls double-exponentially with iterations whenever the degree-2 subgraph of the protograph is cycle-free and noise level is below threshold. Conditions for stability of protograph density evolution are established and related to the structure of the protograph. Using large-girth graphs, sequences of protograph LDPC codes with block-error threshold equal to bit-error threshold and block-error rate falling near-exponentially with blocklength are constructed deterministically. Small-sized protographs are optimized to obtain thresholds near capacity for binary erasure and binary-input Gaussian channels.

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arXiv2015

Approximation of Capacity for ISI Channels with One-bit Output Quantization

Motivated by recent high bandwidth communication systems, Inter-Symbol Interference (ISI) channels with 1-bit quantized output are considered under an average-power-constrained continuous input. While the exact capacity is difficult to characterize, an approximation that matches with the exact channel output up to a probability of error is provided. The approximation does not have additive noise, but constrains the channel output (without noise) to be above a threshold in absolute value. The capacity under the approximation is computed using methods involving standard Gibbs distributions. Markovian achievable schemes approaching the approximate capacity are provided. The methods used over the approximate ISI channel result in ideas for practical coding schemes for ISI channels with 1-bit output quantization.

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arXiv2014

Secure Compute-and-Forward in a Bidirectional Relay

We consider the basic bidirectional relaying problem, in which two users in a wireless network wish to exchange messages through an intermediate relay node. In the compute-and-forward strategy, the relay computes a function of the two messages using the naturally-occurring sum of symbols simultaneously transmitted by user nodes in a Gaussian multiple access (MAC) channel, and the computed function value is forwarded to the user nodes in an ensuing broadcast phase. In this paper, we study the problem under an additional security constraint, which requires that each user's message be kept secure from the relay. We consider two types of security constraints: perfect secrecy, in which the MAC channel output seen by the relay is independent of each user's message; and strong secrecy, which is a form of asymptotic independence. We propose a coding scheme based on nested lattices, the main feature of which is that given a pair of nested lattices that satisfy certain "goodness" properties, we can explicitly specify probability distributions for randomization at the encoders to achieve the desired security criteria. In particular, our coding scheme guarantees perfect or strong secrecy even in the absence of channel noise. The noise in the channel only affects reliability of computation at the relay, and for Gaussian noise, we derive achievable rates for reliable and secure computation. We also present an application of our methods to the multi-hop line network in which a source needs to transmit messages to a destination through a series of intermediate relays.

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arXiv2014

Sub-Modularity of Waterfilling with Applications to Online Basestation Allocation

We show that the popular water-filling algorithm for maximizing the mutual information in parallel Gaussian channels is sub-modular. The sub-modularity of water-filling algorithm is then used to derive online basestation allocation algorithms, where mobile users are assigned to one of many possible basestations immediately and irrevocably upon arrival without knowing the future user information. The goal of the allocation is to maximize the sum-rate of the system under power allocation at each basestation. We present online algorithms with competitive ratio of at most 2 when compared to offline algorithms that have knowledge of all future user arrivals.

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arXiv2013

The Gaussian Two-way Diamond Channel

We consider two-way relaying in a Gaussian diamond channel, where two terminal nodes wish to exchange information using two relays. A simple baseline protocol is obtained by time-sharing between two one-way protocols. To improve upon the baseline performance, we propose two compute-and-forward (CF) protocols: Compute-and-forward Compound multiple access channel (CF-CMAC) and Compute-and-forward-Broadcast (CF-BC). These protocols mix the two flows through the two relays and achieve rates better than the simple time-sharing protocol. We derive an outer bound to the capacity region that is satisfied by any relaying protocol, and observe that the proposed protocols provide rates close to the outer bound in certain channel conditions. Both the CF-CMAC and CF-BC protocols use nested lattice codes in the compute phases. In the CF-CMAC protocol, both relays simultaneously forward to the destinations over a Compound Multiple Access Channel (CMAC). In the simpler CF-BC protocol's forward phase, one relay is selected at a time for Broadcast Channel (BC) transmission depending on the rate-pair to be achieved. We also consider the diamond channel with direct source-destination link and the diamond channel with interfering relays. Outer bounds and achievable rate regions are compared for these two channels as well. Mixing of flows using the CF-CMAC protocol is shown to be good for symmetric two-way rates.

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arXiv2013

Online Algorithms for Basestation Allocation

Design of {\it online algorithms} for assigning mobile users to basestations is considered with the objective of maximizing the sum-rate, when all users associated to any one basestation equally share each basestation's resources. Each user on its arrival reveals the rates it can obtain if connected to each of the basestations, and the problem is to assign each user to any one basestation irrevocably so that the sum-rate is maximized at the end of all user arrivals, without knowing the future user arrival or rate information or its statistics at each user arrival. Online algorithms with constant factor loss in comparison to offline algorithms (that know both the user arrival and user rates profile in advance) are derived. The proposed online algorithms are motivated from the famous online k-secretary problem and online maximum weight matching problem.

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arXiv2013

Deterministic Constructions for Large Girth Protograph LDPC Codes

The bit-error threshold of the standard ensemble of Low Density Parity Check (LDPC) codes is known to be close to capacity, if there is a non-zero fraction of degree-two bit nodes. However, the degree-two bit nodes preclude the possibility of a block-error threshold. Interestingly, LDPC codes constructed using protographs allow the possibility of having both degree-two bit nodes and a block-error threshold. In this paper, we analyze density evolution for protograph LDPC codes over the binary erasure channel and show that their bit-error probability decreases double exponentially with the number of iterations when the erasure probability is below the bit-error threshold and long chain of degree-two variable nodes are avoided in the protograph. We present deterministic constructions of such protograph LDPC codes with girth logarithmic in blocklength, resulting in an exponential fall in bit-error probability below the threshold. We provide optimized protographs, whose block-error thresholds are better than that of the standard ensemble with minimum bit-node degree three. These protograph LDPC codes are theoretically of great interest, and have applications, for instance, in coding with strong secrecy over wiretap channels.

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arXiv2012

Outer Bounds for the Capacity Region of a Gaussian Two-way Relay Channel

We consider a three-node half-duplex Gaussian relay network where two nodes (say $a$, $b$) want to communicate with each other and the third node acts as a relay for this twoway communication. Outer bounds and achievable rate regions for the possible rate pairs $(R_a, R_b)$ for two-way communication are investigated. The modes (transmit or receive) of the halfduplex nodes together specify the state of the network. A relaying protocol uses a specific sequence of states and a coding scheme for each state. In this paper, we first obtain an outer bound for the rate region of all achievable $(R_a,R_b)$ based on the half-duplex cut-set bound. This outer bound can be numerically computed by solving a linear program. It is proved that at any point on the boundary of the outer bound only four of the six states of the network are used. We then compare it with achievable rate regions of various known protocols. We consider two kinds of protocols: (1) protocols in which all messages transmitted in a state are decoded with the received signal in the same state, and (2) protocols where information received in one state can also be stored and used as side information to decode messages in future states. Various conclusions are drawn on the importance of using all states, use of side information, and the choice of processing at the relay. Then, two analytical outer bounds (as opposed to an optimization problem formulation) are derived. Using an analytical outer bound, we obtain the symmetric capacity within 0.5 bits for some channel conditions where the direct link between nodes a and b is weak.

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arXiv2012

Secure Computation in a Bidirectional Relay

Bidirectional relaying, where a relay helps two user nodes to exchange equal length binary messages, has been an active area of recent research. A popular strategy involves a modified Gaussian MAC, where the relay decodes the XOR of the two messages using the naturally-occurring sum of symbols simultaneously transmitted by user nodes. In this work, we consider the Gaussian MAC in bidirectional relaying with an additional secrecy constraint for protection against a honest but curious relay. The constraint is that, while the relay should decode the XOR, it should be fully ignorant of the individual messages of the users. We exploit the symbol addition that occurs in a Gaussian MAC to design explicit strategies that achieve perfect independence between the received symbols and individual transmitted messages. Our results actually hold for a more general scenario where the messages at the two user nodes come from a finite Abelian group, and the relay must decode the sum within the group of the two messages. We provide a lattice coding strategy and study optimal rate versus average power trade-offs for asymptotically large dimensions.

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arXiv2011

The Treewidth of MDS and Reed-Muller Codes

The constraint complexity of a graphical realization of a linear code is the maximum dimension of the local constraint codes in the realization. The treewidth of a linear code is the least constraint complexity of any of its cycle-free graphical realizations. This notion provides a useful parametrization of the maximum-likelihood decoding complexity for linear codes. In this paper, we prove the surprising fact that for maximum distance separable codes and Reed-Muller codes, treewidth equals trelliswidth, which, for a code, is defined to be the least constraint complexity (or branch complexity) of any of its trellis realizations. From this, we obtain exact expressions for the treewidth of these codes, which constitute the only known explicit expressions for the treewidth of algebraic codes.

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arXiv2011

Strong Secrecy on the Binary Erasure Wiretap Channel Using Large-Girth LDPC Codes

For an arbitrary degree distribution pair (DDP), we construct a sequence of low-density parity-check (LDPC) code ensembles with girth growing logarithmically in block-length using Ramanujan graphs. When the DDP has minimum left degree at least three, we show using density evolution analysis that the expected bit-error probability of these ensembles, when passed through a binary erasure channel with erasure probability $ε$, decays as $\mathcal{O}(\exp(-c_1 n^{c_2}))$ with the block-length $n$ for positive constants $c_1$ and $c_2$, as long as $ε$ is lesser than the erasure threshold $ε_\mathrm{th}$ of the DDP. This guarantees that the coset coding scheme using the dual sequence provides strong secrecy over the binary erasure wiretap channel for erasure probabilities greater than $1 - ε_\mathrm{th}$.

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arXiv2010

Path Gain Algebraic Formulation for the Scalar Linear Network Coding Problem

In the algebraic view, the solution to a network coding problem is seen as a variety specified by a system of polynomial equations typically derived by using edge-to-edge gains as variables. The output from each sink is equated to its demand to obtain polynomial equations. In this work, we propose a method to derive the polynomial equations using source-to-sink path gains as the variables. In the path gain formulation, we show that linear and quadratic equations suffice; therefore, network coding becomes equivalent to a system of polynomial equations of maximum degree 2. We present algorithms for generating the equations in the path gains and for converting path gain solutions to edge-to-edge gain solutions. Because of the low degree, simplification is readily possible for the system of equations obtained using path gains. Using small-sized network coding problems, we show that the path gain approach results in simpler equations and determines solvability of the problem in certain cases. On a larger network (with 87 nodes and 161 edges), we show how the path gain approach continues to provide deterministic solutions to some network coding problems.

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arXiv2010

Strong Secrecy for Erasure Wiretap Channels

We show that duals of certain low-density parity-check (LDPC) codes, when used in a standard coset coding scheme, provide strong secrecy over the binary erasure wiretap channel (BEWC). This result hinges on a stopping set analysis of ensembles of LDPC codes with block length $n$ and girth $\geq 2k$, for some $k \geq 2$. We show that if the minimum left degree of the ensemble is $l_\mathrm{min}$, the expected probability of block error is $\calO(\frac{1}{n^{\lceil l_\mathrm{min} k /2 \rceil - k}})$ when the erasure probability $ε< ε_\mathrm{ef}$, where $ε_\mathrm{ef}$ depends on the degree distribution of the ensemble. As long as $l_\mathrm{min} > 2$ and $k > 2$, the dual of this LDPC code provides strong secrecy over a BEWC of erasure probability greater than $1 - ε_\mathrm{ef}$.

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arXiv2010

NLHB : A Non-Linear Hopper Blum Protocol

In this paper, we propose a light-weight provably-secure authentication protocol called the NLHB protocol, which is a variant of the HB protocol. The HB protocol uses the complexity of decoding linear codes for security against passive attacks. In contrast, security for the NLHB protocol is proved by reducing passive attacks to the problem of decoding a class of non-linear codes that are provably hard. We demonstrate that the existing passive attacks on the HB protocol family, which have contributed to considerable reduction in its effective key-size, are ineffective against the NLHB protocol. From the evidence, we conclude that smaller-key sizes are sufficient for the NLHB protocol to achieve the same level of passive attack security as the HB Protocol. Further, for this choice of parameters, we provide an implementation instance for the NLHB protocol for which the Prover/Verifier complexity is lower than the HB protocol, enabling authentication on very low-cost devices like RFID tags. Finally, in the spirit of the HB$^{+}$ protocol, we extend the NLHB protocol to the NLHB$^{+}$ protocol and prove security against the class of active attacks defined in the DET Model.

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arXiv2010

Multistage Relaying Using Interference Networks

Wireless networks with multiple nodes that relay information from a source to a destination are expected to be deployed in many applications. Therefore, understanding their design and performance under practical constraints is important. In this work, we propose and study three multihopping decode and forward (MDF) protocols for multistage half-duplex relay networks with no direct link between the source and destination nodes. In all three protocols, we assume no cooperation across relay nodes for encoding and decoding. Numerical evaluation in illustrative example networks and comparison with cheap relay cut-set bounds for half-duplex networks show that the proposed MDF protocols approach capacity in some ranges of channel gains. The main idea in the design of the protocols is the use of coding in interference networks that are created in different states or modes of a half-duplex network. Our results suggest that multistage half-duplex relaying with practical constraints on cooperation is comparable to point-to-point links and full-duplex relay networks, if there are multiple non-overlapping paths from source to destination and if suitable coding is employed in interference network states.

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arXiv2010

Dirty Paper Coding using Sign-bit Shaping and LDPC Codes

Dirty paper coding (DPC) refers to methods for pre-subtraction of known interference at the transmitter of a multiuser communication system. There are numerous applications for DPC, including coding for broadcast channels. Recently, lattice-based coding techniques have provided several designs for DPC. In lattice-based DPC, there are two codes - a convolutional code that defines a lattice used for shaping and an error correction code used for channel coding. Several specific designs have been reported in the recent literature using convolutional and graph-based codes for capacity-approaching shaping and coding gains. In most of the reported designs, either the encoder works on a joint trellis of shaping and channel codes or the decoder requires iterations between the shaping and channel decoders. This results in high complexity of implementation. In this work, we present a lattice-based DPC scheme that provides good shaping and coding gains with moderate complexity at both the encoder and the decoder. We use a convolutional code for sign-bit shaping, and a low-density parity check (LDPC) code for channel coding. The crucial idea is the introduction of a one-codeword delay and careful parsing of the bits at the transmitter, which enable an LDPC decoder to be run first at the receiver. This provides gains without the need for iterations between the shaping and channel decoders. Simulation results confirm that at high rates the proposed DPC method performs close to capacity with moderate complexity. As an application of the proposed DPC method, we show a design for superposition coding that provides rates better than time-sharing over a Gaussian broadcast channel.

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arXiv2009

Codes on Planar Graphs

Codes defined on graphs and their properties have been subjects of intense recent research. On the practical side, constructions for capacity-approaching codes are graphical. On the theoretical side, codes on graphs provide several intriguing problems in the intersection of coding theory and graph theory. In this paper, we study codes defined by planar Tanner graphs. We derive an upper bound on minimum distance $d$ of such codes as a function of the code rate $R$ for $R \ge 5/8$. The bound is given by $$d\le \lceil \frac{7-8R}{2(2R-1)} \rceil + 3\le 7.$$ Among the interesting conclusions of this result are the following: (1) planar graphs do not support asymptotically good codes, and (2) finite-length, high-rate codes on graphs with high minimum distance will necessarily be non-planar.

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arXiv2008

Reed-Solomon Subcodes with Nontrivial Traces: Distance Properties and Soft-Decision Decoding

Reed-Solomon (RS) codes over GF$(2^m)$ have traditionally been the most popular non-binary codes in almost all practical applications. The distance properties of RS codes result in excellent performance under hard-decision bounded-distance decoding. However, efficient and implementable soft decoding for high-rate (about 0.9) RS codes over large fields (GF(256), say) continues to remain a subject of research with a promise of further coding gains. In this work, our objective is to propose and investigate $2^m$-ary codes with non-trivial binary trace codes as an alternative to RS codes. We derive bounds on the rate of a $2^m$-ary code with a non-trivial binary trace code. Then we construct certain subcodes of RS codes over GF($2^m$) that have a non-trivial binary trace with distances and rates meeting the derived bounds. The properties of these subcodes are studied and low-complexity hard-decision and soft-decision decoders are proposed. The decoders are analyzed, and their performance is compared with that of comparable RS codes. Our results suggest that these subcodes of RS codes could be viable alternatives for RS codes in applications.

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