21 research outputs found
Lossy Source Coding via Spatially Coupled LDGM Ensembles
We study a new encoding scheme for lossy source compression based on
spatially coupled low-density generator-matrix codes. We develop a
belief-propagation guided-decimation algorithm, and show that this algorithm
allows to approach the optimal distortion of spatially coupled ensembles.
Moreover, using the survey propagation formalism, we also observe that the
optimal distortions of the spatially coupled and individual code ensembles are
the same. Since regular low-density generator-matrix codes are known to achieve
the Shannon rate-distortion bound under optimal encoding as the degrees grow,
our results suggest that spatial coupling can be used to reach the
rate-distortion bound, under a {\it low complexity} belief-propagation
guided-decimation algorithm.
This problem is analogous to the MAX-XORSAT problem in computer science.Comment: Submitted to ISIT 201
Approaching the Rate-Distortion Limit with Spatial Coupling, Belief propagation and Decimation
We investigate an encoding scheme for lossy compression of a binary symmetric
source based on simple spatially coupled Low-Density Generator-Matrix codes.
The degree of the check nodes is regular and the one of code-bits is Poisson
distributed with an average depending on the compression rate. The performance
of a low complexity Belief Propagation Guided Decimation algorithm is
excellent. The algorithmic rate-distortion curve approaches the optimal curve
of the ensemble as the width of the coupling window grows. Moreover, as the
check degree grows both curves approach the ultimate Shannon rate-distortion
limit. The Belief Propagation Guided Decimation encoder is based on the
posterior measure of a binary symmetric test-channel. This measure can be
interpreted as a random Gibbs measure at a "temperature" directly related to
the "noise level of the test-channel". We investigate the links between the
algorithmic performance of the Belief Propagation Guided Decimation encoder and
the phase diagram of this Gibbs measure. The phase diagram is investigated
thanks to the cavity method of spin glass theory which predicts a number of
phase transition thresholds. In particular the dynamical and condensation
"phase transition temperatures" (equivalently test-channel noise thresholds)
are computed. We observe that: (i) the dynamical temperature of the spatially
coupled construction saturates towards the condensation temperature; (ii) for
large degrees the condensation temperature approaches the temperature (i.e.
noise level) related to the information theoretic Shannon test-channel noise
parameter of rate-distortion theory. This provides heuristic insight into the
excellent performance of the Belief Propagation Guided Decimation algorithm.
The paper contains an introduction to the cavity method
How to Achieve the Capacity of Asymmetric Channels
We survey coding techniques that enable reliable transmission at rates that
approach the capacity of an arbitrary discrete memoryless channel. In
particular, we take the point of view of modern coding theory and discuss how
recent advances in coding for symmetric channels help provide more efficient
solutions for the asymmetric case. We consider, in more detail, three basic
coding paradigms.
The first one is Gallager's scheme that consists of concatenating a linear
code with a non-linear mapping so that the input distribution can be
appropriately shaped. We explicitly show that both polar codes and spatially
coupled codes can be employed in this scenario. Furthermore, we derive a
scaling law between the gap to capacity, the cardinality of the input and
output alphabets, and the required size of the mapper.
The second one is an integrated scheme in which the code is used both for
source coding, in order to create codewords distributed according to the
capacity-achieving input distribution, and for channel coding, in order to
provide error protection. Such a technique has been recently introduced by
Honda and Yamamoto in the context of polar codes, and we show how to apply it
also to the design of sparse graph codes.
The third paradigm is based on an idea of B\"ocherer and Mathar, and
separates the two tasks of source coding and channel coding by a chaining
construction that binds together several codewords. We present conditions for
the source code and the channel code, and we describe how to combine any source
code with any channel code that fulfill those conditions, in order to provide
capacity-achieving schemes for asymmetric channels. In particular, we show that
polar codes, spatially coupled codes, and homophonic codes are suitable as
basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published
in IEEE Trans. Inform. Theor
On privacy amplification, lossy compression, and their duality to channel coding
We examine the task of privacy amplification from information-theoretic and
coding-theoretic points of view. In the former, we give a one-shot
characterization of the optimal rate of privacy amplification against classical
adversaries in terms of the optimal type-II error in asymmetric hypothesis
testing. This formulation can be easily computed to give finite-blocklength
bounds and turns out to be equivalent to smooth min-entropy bounds by Renner
and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a
bound in terms of the divergence by Yang, Schaefer, and Poor
[arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy
amplification based on linear codes can be easily repurposed for channel
simulation. Combined with known relations between channel simulation and lossy
source coding, this implies that privacy amplification can be understood as a
basic primitive for both channel simulation and lossy compression. Applied to
symmetric channels or lossy compression settings, our construction leads to
proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing
to the notion of channel duality recently detailed by us in [IEEE Trans. Info.
Theory 64, 577 (2018)], we show that linear error-correcting codes for
symmetric channels with quantum output can be transformed into linear lossy
source coding schemes for classical variables arising from the dual channel.
This explains a "curious duality" in these problems for the (self-dual) erasure
channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and
partly anticipates recent results on optimal lossy compression by polar and
low-density generator matrix codes.Comment: v3: updated to include equivalence of the converse bound with smooth
entropy formulations. v2: updated to include comparison with the one-shot
bounds of arXiv:1706.03866. v1: 11 pages, 4 figure
Capacity-Achieving Coding Mechanisms: Spatial Coupling and Group Symmetries
The broad theme of this work is in constructing optimal transmission mechanisms for a wide variety of communication systems. In particular, this dissertation provides a proof of threshold saturation for spatially-coupled codes, low-complexity capacity-achieving coding schemes for side-information problems, a proof that Reed-Muller and primitive narrow-sense BCH codes achieve capacity on erasure channels, and a mathematical framework to design delay sensitive communication systems.
Spatially-coupled codes are a class of codes on graphs that are shown to achieve capacity universally over binary symmetric memoryless channels (BMS) under belief-propagation decoder. The underlying phenomenon behind spatial coupling, known as “threshold saturation via spatial coupling”, turns out to be general and this technique has been applied to a wide variety of systems. In this work, a proof of the threshold saturation phenomenon is provided for irregular low-density parity-check (LDPC) and low-density generator-matrix (LDGM) ensembles on BMS channels. This proof is far simpler than published alternative proofs and it remains as the only technique to handle irregular and LDGM codes. Also, low-complexity capacity-achieving codes are constructed for three coding problems via spatial coupling: 1) rate distortion with side-information, 2) channel coding with side-information, and 3) write-once memory system. All these schemes are based on spatially coupling compound LDGM/LDPC ensembles.
Reed-Muller and Bose-Chaudhuri-Hocquengham (BCH) are well-known algebraic codes introduced more than 50 years ago. While these codes are studied extensively in the literature it wasn’t known whether these codes achieve capacity. This work introduces a technique to show that Reed-Muller and primitive narrow-sense BCH codes achieve capacity on erasure channels under maximum a posteriori (MAP) decoding. Instead of relying on the weight enumerators or other precise details of these codes, this technique requires that these codes have highly symmetric permutation groups. In fact, any sequence of linear codes with increasing blocklengths whose rates converge to a number between 0 and 1, and whose permutation groups are doubly transitive achieve capacity on erasure channels under bit-MAP decoding. This pro-vides a rare example in information theory where symmetry alone is sufficient to achieve capacity.
While the channel capacity provides a useful benchmark for practical design, communication systems of the day also demand small latency and other link layer metrics. Such delay sensitive communication systems are studied in this work, where a mathematical framework is developed to provide insights into the optimal design of these systems
저밀도 부호의 응용: 묶음 지그재그 파운틴 부호와 WOM 부호
학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2017. 2. 노종선.This dissertation contains the following two contributions on the applications of sparse codes.
Fountain codes Batched zigzag (BZ) fountain codes
– Two-phase batched zigzag (TBZ) fountain codes
Write-once memory (WOM) codes
– WOM codes implemented by rate-compatible low-density generator matrix (RC-LDGM) codes
First, two classes of fountain codes, called batched zigzag fountain codes and two-phase batched zigzag fountain codes, are proposed for the symbol erasure channel. At a cost of slightly lengthened code symbols, the involved message symbols in each batch of the proposed codes can be recovered by low complexity zigzag decoding algorithm. Thus, the proposed codes have low buffer occupancy during decoding process. These features are suitable for receivers with limited hardware resources in the broadcasting channel. A method to obtain degree distributions of code symbols for the proposed codes via ripple size evolution is also proposed by taking into account the released code symbols from the batches. It is shown that the proposed codes outperform Luby transform codes and zigzag decodable fountain codes with respect to intermediate recovery rate and coding overhead when message length is short, symbol erasure rate is low, and available buffer size is limited.
In the second part of this dissertation, WOM codes constructed by sparse codes are presented. Recently, WOM codes are adopted to NAND flash-based solid-state drive (SSD) in order to extend the lifetime by reducing the number of erasure operations. Here, a new rewriting scheme for the SSD is proposed, which is implemented by multiple binary erasure quantization (BEQ) codes. The corresponding BEQ codes are constructed by RC-LDGM codes. Moreover, by putting RC-LDGM codes together with a page selection method, writing efficiency can be improved. It is verified via simulation that the SSD with proposed rewriting scheme outperforms the SSD without and with the conventional WOM codes for single level cell (SLC) and multi-level cell (MLC) flash memories.1 Introduction 1
1.1 Background 1
1.2 Overview of Dissertation 5
2 Sparse Codes 7
2.1 Linear Block Codes 7
2.2 LDPC Codes 9
2.3 Message Passing Decoder 11
3 New Fountain Codes with Improved Intermediate Recovery Based on Batched Zigzag Coding 13
3.1 Preliminaries 17
3.1.1 Definitions and Notation 17
3.1.2 LT Codes 18
3.1.3 Zigzag Decodable Codes 20
3.1.4 Bit-Level Overhead 22
3.2 New Fountain Codes Based on Batched Zigzag Coding 23
3.2.1 Construction of Shift Matrix 24
3.2.2 Encoding and Decoding of the Proposed BZ Fountain Codes 25
3.2.3 Storage and Computational Complexity 28
3.3 Degree Distribution of BZ Fountain Codes 31
3.3.1 Relation Between and 31
3.3.2 Derivation of via Ripple Size Evolution 32
3.4 Two-Phase Batched Zigzag Fountain Codes with Additional Memory 40
3.4.1 Code Construction 41
3.4.2 Bit-Level Overhead 46
3.5 Numerical Analysis 49
4 Write-Once Memory Codes Using Rate-Compatible LDGM Codes 60
4.1 Preliminaries 62
4.1.1 NAND Flash Memory 62
4.1.2 Rewriting Schemes for Flash Memory 62
4.1.3 Construction of Rewriting Codes by BEQ Codes 65
4.2 Proposed Rewriting Codes 67
4.2.1 System Model 67
4.2.2 Multi-rate Rewriting Codes 68
4.2.3 Page Selection for Rewriting 70
4.3 RC-LDGM Codes 74
4.4 Numerical Analysis 76
5 Conclusions 80
Bibliography 82
초록 94Docto