171 research outputs found
Entanglement-assisted quantum low-density parity-check codes
This paper develops a general method for constructing entanglement-assisted
quantum low-density parity-check (LDPC) codes, which is based on combinatorial
design theory. Explicit constructions are given for entanglement-assisted
quantum error-correcting codes (EAQECCs) with many desirable properties. These
properties include the requirement of only one initial entanglement bit, high
error correction performance, high rates, and low decoding complexity. The
proposed method produces infinitely many new codes with a wide variety of
parameters and entanglement requirements. Our framework encompasses various
codes including the previously known entanglement-assisted quantum LDPC codes
having the best error correction performance and many new codes with better
block error rates in simulations over the depolarizing channel. We also
determine important parameters of several well-known classes of quantum and
classical LDPC codes for previously unsettled cases.Comment: 20 pages, 5 figures. Final version appearing in Physical Review
New Protograph-Based Construction of GLDPC Codes for Binary Erasure Channel and LDPC Codes for Block Fading Channel
ํ์๋
ผ๋ฌธ(๋ฐ์ฌ) -- ์์ธ๋ํ๊ต๋ํ์ : ๊ณต๊ณผ๋ํ ์ ๊ธฐยท์ ๋ณด๊ณตํ๋ถ, 2022.2. ๋
ธ์ข
์ ๊ต์๋.์ด ํ์ ๋
ผ๋ฌธ์์๋ ๋ค์ ๋ ๊ฐ์ง์ ์ฐ๊ตฌ๊ฐ ์ด๋ฃจ์ด์ก๋ค: i) ์ด์ง ์์ค ์ฑ๋์์ ์๋ก์ด ๊ตฌ์กฐ์ ํ๋กํ ๊ทธ๋ํ ๊ธฐ๋ฐ generalized low-density parity-check (GLDPC) ๋ถํธ์ ์ค๊ณ ๋ฐฉ๋ฒ ii) ๋ธ๋ก ํ์ด๋ฉ ์ฑ๋์ ์ํ ํ๋กํ ๊ทธ๋ํ ๊ธฐ๋ฐ์ LDPC ๋ถํธ ์ค๊ณ.
์ฒซ ๋ฒ์งธ๋ก, ์ด์ง ์์ค ์ฑ๋์์ ์๋กญ๊ฒ ์ ์๋ ๋ถ๋ถ์ ๋ํ ๊ธฐ๋ฒ์ ์ด์ฉํ ํ๋กํ ๊ทธ๋ํ ๊ธฐ๋ฐ์ GLDPC ๋ถํธ๊ฐ ์ ์๋์๋ค. ๊ธฐ์กด์ ํ๋กํ ๊ทธ๋ํ ๊ธฐ๋ฐ์ GLDPC ๋ถํธ์ ๊ฒฝ์ฐ ํ๋กํ ๊ทธ๋ํ ์์ญ์์ single parity-check (SPC) ๋
ธ๋๋ฅผ generalized constraint (GC) ๋
ธ๋๋ก ์นํ(๋ํ)ํ๋ ํํ๋ก ๋ถํธ๊ฐ ์ค๊ณ๋์ด ์ฌ๋ฌ ๋ณ์ ๋
ธ๋ ๊ฑธ์ณ GC ๋
ธ๋๊ฐ ์ฐ๊ฒฐ๋๋ ํํ๋ฅผ ๊ฐ์ง๋ค. ๋ฐ๋ฉด, ์ ์๋ ๋ถ๋ถ์ ๋ํ ๊ธฐ๋ฒ์ ํ ๊ฐ์ ๋ณ์ ๋
ธ๋์ GC ๋
ธ๋๋ฅผ ์ฐ๊ฒฐํ๋๋ก ๋ง๋ค ์ ์๋ค. ๋ฐ๊ฟ ๋งํ๋ฉด, ์ ์๋ ๋ถ๋ถ์ ๋ํ ๊ธฐ๋ฒ์ ๋ ์ธ๋ฐํ ๋ํ์ด ๊ฐ๋ฅํด์ ๊ฒฐ๊ณผ์ ์ผ๋ก ๋ถํธ ์ค๊ณ์ ์์ด ๋์ ์์ ๋๋ฅผ ๊ฐ์ง๊ณ ๋ ์ธ๋ จ๋ ๋ถํธ ์ต์ ํ๊ฐ ๊ฐ๋ฅํ๋ค. ๋ณธ ํ์ ๋
ผ๋ฌธ์์๋ ๋ถ๋ถ์ ๋ํ๊ณผ PEXIT ๋ถ์์ ์ด์ฉํ์ฌ partially doped GLDPC (PD-GLDPC) ๋ถํธ๋ฅผ ์ค๊ณํ๊ณ ์ต์ ํ ํ์๋ค. ๋๋ถ์ด, PD-GLDPC ๋ถํธ์ ์ผ๋ฐ์ ์ต์ ๊ฑฐ๋ฆฌ๋ฅผ ๊ฐ์ง๋ ์กฐ๊ฑด์ ์ ์ํ์๊ณ ์ด๋ฅผ ์ด
๋ก ์ ์ผ๋ก ์ฆ๋ช
ํ์๋ค. ๊ฒฐ๊ณผ์ ์ผ๋ก, ์ ์๋ PD-GLDPC ๋ถํธ๋ ํ์กดํ๋ GLDPC ๋ถํธ์ ์ฑ๋ฅ๋ณด๋ค ์ ์๋ฏธํ๊ฒ ์ํฐํ ์ฑ๋ฅ์ด ์ข์๊ณ ๋์์ ์ค๋ฅ ๋ง๋ฃจ๊ฐ ์์๋ค. ๋ง์ง๋ง์ผ๋ก, ์ต์ ํ๋ PD-GLDPC ๋ถํธ๋ ํ์กดํ๋ ์ต์ ๋ธ๋ก LDPC ๋ถํธ๋ค์ ๊ทผ์ ํ ์ฑ๋ฅ์ ๊ฐ์ง์ ๋ณด์ฌ์ฃผ์๋ค.
๋ ๋ฒ์งธ๋ก, ๋ธ๋ก ํ์ด๋ฉ (BF) ์ฑ๋์์ resolvable block design (RBD)๋ฅผ ์ด์ฉํ ํ๋กํ ๊ทธ๋ํ LDPC ๋ถํธ ์ค๊ณ๊ฐ ์ด๋ฃจ์ด์ก๋ค. ์ ์๋ ๋ถํธ์ ์ฑ๋ฅ์ ํ์ธํ๊ธฐ ์ํ ๋นํธ ์ค๋ฅ์จ์ ์ํ์ ๊ฐ๋ง ์งํ๋ผ๋ ์ ์๋ ๊ธฐ๋ฒ์ ์ด์ฉํด ์ ๋ํ์๋ค. ๋ํ, ์๋ฎฌ๋ ์ด์
์ ํตํด ์ ๋๋ ์ค๋ฅ์จ ์ํ๊ณผ ๋ถํธ์ ํ๋ ์ ์ค๋ฅ์จ์ด ๋์ SNR ์์ญ์์ ์ฑ๋ outage ํ๋ฅ ์ ๊ทผ์ ํจ์ ์ ์ ์๋ค.In this dissertation, two main contributions are given as: i) new construction methods for protograph-based generalized low-density parity-check (GLDPC) codes for the binary erasure channel using partial doping technique and ii) new design of protograph-based low-density parity-check (LDPC) codes for the block fading channel using resolvable block design.
First, a new code design technique, called partial doping, for protograph-based GLDPC codes is proposed. While the conventional construction method of protograph-based GLDPC codes is to replace some single parity-check (SPC) nodes with generalized constraint (GC) nodes applying to multiple connected variable nodes (VNs) in the protograph, the proposed technique of partial doping can select any number of partial VNs in the protograph to be protected by GC nodes.
In other words, the partial doping technique enables finer tuning of doping, which gives higher degrees of freedom in the code design and enables a sophisticated code optimization. The proposed partially doped GLDPC (PD-GLDPC) codes are constructed using the partial doping technique and optimized by the protograph extrinsic information transfer (PEXIT) analysis.
In addition, the condition guaranteeing the linear minimum distance growth of the PD-GLDPC codes is proposed and analytically proven so that the PD-GLDPC code ensembles satisfying this condition have the typical minimum distance.
Consequently, the proposed PD-GLDPC codes outperform the conventional GLDPC codes with a notable improvement in the waterfall performance and without the error floor phenomenon.
Finally, the PD-GLDPC codes are shown to achieve a competitive performance compared to the existing state-of-the-art block LDPC codes.
Second, the protograph-based LDPC codes constructed from resolvable balanced incomplete block design (RBIBD) are designed and proposed for block fading (BF) channel.
In order to analyze the performance of the proposed LDPC codes, the upper bounds on bit error rate (BER) using the novel method called gamma evolution are derived.
In addition, the numerical analysis shows that the upper bound and the frame error rate (FER) of the proposed LDPC codes approach the channel outage probability in a finite signal-to-noise ratio (SNR) region.1 INTRODUCTION 1
1.1 Background 1
1.2 Overview of Dissertation 3
2 Overview of LDPC Codes 5
2.1 LDPC Codes 5
2.2 Decoding of LDPC Codes in the BEC 7
2.3 Analysis tool for LDPC Codes 8
2.3.1 Density Evolution 8
2.4 Protograph-Based LDPC Codes 9
3 Construction of Protograph-Based Partially Doped Generalized LDPC Codes 11
3.1 Code Structure of Protograph-Based GLDPC Ensembles 14
3.1.1 Construction of Protograph Doped GLDPC Codes 14
3.1.2 PEXIT Analysis and Decoding Process of Protograph Doped GLDPC Codes 15
3.2 The Proposed PD-GLDPC Codes 18
3.2.1 Construction Method of PD-GLDPC Codes 18
3.2.2 PEXIT Analysis of PD-GLDPC Codes 22
3.2.3 Condition for the Existence of the Typical Minimum Distance of the PD-GLDPC Code Ensemble 23
3.2.4 Comparison between Proposed PD-GLDPC Codes and Protograph Doped GLDPC Codes 25
3.3 Optimization of PD-GLDPC Codes 26
3.3.1 Construction of PD-GLDPC Codes from Regular Protographs 26
3.3.2 Differential Evolution-Based Code Construction from the Degree Distribution of Random LDPC Code Ensembles 28
3.3.3 Optimization of PD-GLDPC Codes Using Protograph Differential Evolution 32
3.4 Numerical Results and Analysis 36
3.4.1 Simulation Result for Optimized PD-GLDPC Code from Regular and Irregular Random LDPC Code Ensembles 36
3.4.2 Simulation Result for PD-GLDPC Code from Optimized Protograph 43
3.5 Proof of Theorem 1: The Constraint for the Existence of the Typical Minimum Distance of the Proposed Protograph-Based PD-GLDPC Codes 45
4 Design of Protograph-Based LDPC Code Using Resolvable Block Design for Block Fading Channel 52
4.1 Problem Formulation 53
4.1.1 BF Channel Model 53
4.1.2 Performance Metrics of BF Channel 54
4.1.3 Protograph-Based LDPC Codes and QC LDPC Codes 57
4.2 New Construction of Protograph-Based LDPC Codes from Resolvable Block Designs 57
4.2.1 Resolvable Block Designs 57
4.2.2 Construction of the Proposed Protograph-Based LDPC Codes 59
4.2.3 Theoretical Analysis of the Proposed Protograph-Based LDPC Code from RBD 61
4.2.4 Numerical Analysis of the Proposed Protograph-Based LDPC Code Codes for BF Channel 65
4.2.5 BER Comparison with Analytical Results from Gamma Evolution 65
4.2.6 FER Comparison with Channel Outage Probability 67
5 Conclusions 69
Abstract (In Korean) 78๋ฐ
On generalized LDPC codes for ultra reliable communication
Ultra reliable low latency communication (URLLC) is an important feature in
future mobile communication systems, as they will require high data rates, large
system capacity and massive device connectivity [11]. To meet such stringent
requirements, many error-correction codes (ECC)s are being investigated; turbo
codes, low density parity check (LDPC) codes, polar codes and convolutional codes
[70, 92, 38], among many others. In this work, we present generalized low density
parity check (GLDPC) codes as a promising candidate for URLLC.
Our proposal is based on a novel class of GLDPC code ensembles, for which
new analysis tools are proposed. We analyze the trade-o_ between coding rate and
asymptotic performance of a class of GLDPC codes constructed by including a
certain fraction of generalized constraint (GC) nodes in the graph. To incorporate
both bounded distance (BD) and maximum likelihood (ML) decoding at GC nodes
into our analysis without resorting to multi-edge type of degree distribution (DD)s,
we propose the probabilistic peeling decoding (P-PD) algorithm, which models the
decoding step at every GC node as an instance of a Bernoulli random variable with
a successful decoding probability that depends on both the GC block code as well
as its decoding algorithm. The P-PD asymptotic performance over the BEC can
be efficiently predicted using standard techniques for LDPC codes such as Density
evolution (DE) or the differential equation method. We demonstrate that the
simulated P-PD performance accurately predicts the actual performance of the
GLPDC code under ML decoding at GC nodes. We illustrate our analysis for
GLDPC code ensembles with regular and irregular DDs.
This design methodology is applied to construct practical codes for URLLC.
To this end, we incorporate to our analysis the use of quasi-cyclic (QC) structures,
to mitigate the code error floor and facilitate the code very large scale integration
(VLSI) implementation. Furthermore, for the additive white Gaussian noise
(AWGN) channel, we analyze the complexity and performance of the message
passing decoder with various update rules (including standard full-precision sum product and min-sum algorithms) and quantization schemes. The block error rate
(BLER) performance of the proposed GLDPC codes, combined with a complementary
outer code, is shown to outperform a variety of state-of-the-art codes, for
URLLC, including LDPC codes, polar codes, turbo codes and convolutional codes,
at similar complexity rates.Programa Oficial de Doctorado en Multimedia y ComunicacionesPresidente: Juan Josรฉ Murillo Fuentes.- Secretario: Matilde Pilar Sรกnchez Fernรกndez.- Vocal: Javier Valls Coquilla
Structural Design and Analysis of Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes
The discovery of two fundamental error-correcting code families, known as turbo codes and low-density parity-check (LDPC) codes, has led to a revolution in coding theory and to a paradigm shift from traditional algebraic codes towards modern graph-based codes that can be decoded by iterative message passing algorithms.
From then on, it has become a focal point of research to develop powerful LDPC and turbo-like codes.
Besides the classical domain of randomly constructed codes, an alternative and competitive line of research is concerned with highly structured LDPC and turbo-like codes based on combinatorial designs.
Such codes are typically characterized by high code rates already at small to moderate code lengths and good code properties such as the avoidance of harmful 4-cycles in the code's factor graph.
Furthermore, their structure can usually be exploited for an efficient implementation, in particular, they can be encoded with low complexity as opposed to random-like codes. Hence, these codes are suitable for high-speed applications such as magnetic recording or optical communication.
This thesis greatly contributes to the field of structured LDPC codes and systematic repeat-accumulate (sRA) codes as a subclass of turbo-like codes by presenting new combinatorial construction techniques and algebraic methods for an improved code design.
More specifically, novel and infinite families of high-rate structured LDPC codes and sRA codes are presented based on balanced incomplete block designs (BIBDs), which form a subclass of combinatorial designs. Besides of showing excellent error-correcting capabilites under iterative decoding, these codes can be implemented efficiently, since their inner structure enables low-complexity encoding and accelerated decoding algorithms.
A further infinite series of structured LDPC codes is presented based on the notion of transversal designs, which form another subclass of combinatorial designs. By a proper configuration of these codes, they reveal an excellent decoding performance under iterative decoding, in particular, with very low error-floors.
The approach for lowering these error-floors is threefold. First, a thorough analysis of the decoding failures is carried out, resulting in an extensive classification of so-called stopping sets and absorbing sets. These combinatorial entities are known to be the main cause of decoding failures in the error-floor region over the binary erasure channel (BEC) and additive white Gaussian noise (AWGN) channel, respectively. Second, the specific code structures are exploited in order to calculate conditions for the avoidance of the most harmful stopping and absorbing sets. Third, powerful design strategies are derived for the identification of those code instances with the best error-floor performances.
The resulting codes can additionally be encoded with low complexity and thus are ideally suited for practical high-speed applications.
Further investigations are carried out on the infinite family of structured LDPC codes based on finite geometries. It is known that these codes perform very well under iterative decoding and that their encoding can be achieved with low complexity. By combining the latest findings in the fields of finite geometries and combinatorial designs, we generate new theoretical insights about the decoding failures of such codes under iterative decoding. These examinations finally help to identify the geometric codes with the most beneficial error-correcting capabilities over the BEC
CROSSTALK-RESILIANT CODING FOR HIGH DENSITY DIGITAL RECORDING
Increasing the track density in magnetic systems is very difficult due to inter-track interference
(ITI) caused by the magnetic field of adjacent tracks. This work presents a
two-track partial response class 4 magnetic channel with linear and symmetrical ITI; and
explores modulation codes, signal processing methods and error correction codes in order
to mitigate the effects of ITI.
Recording codes were investigated, and a new class of two-dimensional run-length
limited recording codes is described. The new class of codes controls the type of ITI
and has been found to be about 10% more resilient to ITI compared to conventional
run-length limited codes. A new adaptive trellis has also been described that adaptively
solves for the effect of ITI. This has been found to give gains up to 5dB in signal to noise
ratio (SNR) at 40% ITI. It was also found that the new class of codes were about 10%
more resilient to ITI compared to conventional recording codes when decoded with the
new trellis.
Error correction coding methods were applied, and the use of Low Density Parity
Check (LDPC) codes was investigated. It was found that at high SNR, conventional
codes could perform as well as the new modulation codes in a combined modulation and
error correction coding scheme. Results suggest that high rate LDPC codes can mitigate
the effect of ITI, however the decoders have convergence problems beyond 30% ITI
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