Stopping Set Distributions of Some Linear Codes

Stopping sets and stopping set distribution of an low-density parity-check code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let $C$ be a binary $[n,k]$ linear code with parity-check matrix $H$, where the rows of $H$ may be dependent. A stopping set $S$ of $C$ with parity-check matrix $H$ is a subset of column indices of $H$ such that the restriction of $H$ to $S$ does not contain a row of weight one. The stopping set distribution $\{T_i(H)\}_{i=0}^n$ enumerates the number of stopping sets with size $i$ of $C$ with parity-check matrix $H$. Note that stopping sets and stopping set distribution are related to the parity-check matrix $H$ of $C$. Let $H^{*}$ be the parity-check matrix of $C$ which is formed by all the non-zero codewords of its dual code $C^{\perp}$. A parity-check matrix $H$ is called BEC-optimal if $T_i(H)=T_i(H^*), i=0,1,..., n$ and $H$ has the smallest number of rows. On the BEC, iterative decoder of $C$ with BEC-optimal parity-check matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes and the extended Hamming codes.Comment: 33 pages, submitted to IEEE Trans. Inform. Theory, Feb. 201

Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes

We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on Information Theor

Distance Properties of Short LDPC Codes and their Impact on the BP, ML and Near-ML Decoding Performance

Parameters of LDPC codes, such as minimum distance, stopping distance, stopping redundancy, girth of the Tanner graph, and their influence on the frame error rate performance of the BP, ML and near-ML decoding over a BEC and an AWGN channel are studied. Both random and structured LDPC codes are considered. In particular, the BP decoding is applied to the code parity-check matrices with an increasing number of redundant rows, and the convergence of the performance to that of the ML decoding is analyzed. A comparison of the simulated BP, ML, and near-ML performance with the improved theoretical bounds on the error probability based on the exact weight spectrum coefficients and the exact stopping size spectrum coefficients is presented. It is observed that decoding performance very close to the ML decoding performance can be achieved with a relatively small number of redundant rows for some codes, for both the BEC and the AWGN channels

Doped Fountain Coding for Minimum Delay Data Collection in Circular Networks

This paper studies decentralized, Fountain and network-coding based strategies for facilitating data collection in circular wireless sensor networks, which rely on the stochastic diversity of data storage. The goal is to allow for a reduced delay collection by a data collector who accesses the network at a random position and random time. Data dissemination is performed by a set of relays which form a circular route to exchange source packets. The storage nodes within the transmission range of the route's relays linearly combine and store overheard relay transmissions using random decentralized strategies. An intelligent data collector first collects a minimum set of coded packets from a subset of storage nodes in its proximity, which might be sufficient for recovering the original packets and, by using a message-passing decoder, attempts recovering all original source packets from this set. Whenever the decoder stalls, the source packet which restarts decoding is polled/doped from its original source node. The random-walk-based analysis of the decoding/doping process furnishes the collection delay analysis with a prediction on the number of required doped packets. The number of doped packets can be surprisingly small when employed with an Ideal Soliton code degree distribution and, hence, the doping strategy may have the least collection delay when the density of source nodes is sufficiently large. Furthermore, we demonstrate that network coding makes dissemination more efficient at the expense of a larger collection delay. Not surprisingly, a circular network allows for a significantly more (analytically and otherwise) tractable strategies relative to a network whose model is a random geometric graph

새로운 소실 채널을 위한 자기동형 군 복호기 및 부분 접속 복구 부호 및 일반화된 근 프로토그래프 LDPC 부호의 설계

학위논문 (박사)-- 서울대학교 대학원 : 공과대학 전기·컴퓨터공학부, 2019. 2. 노종선.In this dissertation, three main contributions are given asi) new two-stage automorphism group decoders (AGD) for cyclic codes in the erasure channel, ii) new constructions of binary and ternary locally repairable codes (LRCs) using cyclic codes and existing LRCs, and iii) new constructions of high-rate generalized root protograph (GRP) low-density parity-check (LDPC) codes for a nonergodic block interference and partially regular (PR) LDPC codes for follower noise jamming (FNJ), are considered. First, I propose a new two-stage AGD (TS-AGD) for cyclic codes in the erasure channel. Recently, error correcting codes in the erasure channel have drawn great attention for various applications such as distributed storage systems and wireless sensor networks, but many of their decoding algorithms are not practical because they have higher decoding complexity and longer delay. Thus, the AGD for cyclic codes in the erasure channel was introduced, which has good erasure decoding performance with low decoding complexity. In this research, I propose new TS-AGDs for cyclic codes in the erasure channel by modifying the parity check matrix and introducing the preprocessing stage to the AGD scheme. The proposed TS-AGD is analyzed for the perfect codes, BCH codes, and maximum distance separable (MDS) codes. Through numerical analysis, it is shown that the proposed decoding algorithm has good erasure decoding performance with lower decoding complexity than the conventional AGD. For some cyclic codes, it is shown that the proposed TS-AGD achieves the perfect decoding in the erasure channel, that is, the same decoding performance as the maximum likelihood (ML) decoder. For MDS codes, TS-AGDs with the expanded parity check matrix and the submatrix inversion are also proposed and analyzed. Second, I propose new constructions of binary and ternary LRCs using cyclic codes and existing two LRCs for distributed storage system. For a primitive work, new constructions of binary and ternary LRCs using cyclic codes and their concatenation are proposed. Some of proposed binary LRCs with Hamming weights 4, 5, and 6 are optimal in terms of the upper bounds. In addition, the similar method of the binary case is applied to construct the ternary LRCs with good parameters. Also, new constructions of binary LRCs with large Hamming distance and disjoint repair groups are proposed. The proposed binary linear LRCs constructed by using existing binary LRCs are optimal or near-optimal in terms of the bound with disjoint repair group. Last, I propose new constructions of high-rate GRP LDPC codes for a nonergodic block interference and anti-jamming PR LDPC codes for follower jamming. The proposed high-rate GRP LDPC codes are based on nonergodic two-state binary symmetric channel with block interference and Nakagami-$m$ block fading. In these channel environments, GRP LDPC codes have good performance approaching to the theoretical limit in the channel with one block interference, where their performance is shown by the channel threshold or the channel outage probability. In the proposed design, I find base matrices using the protograph extrinsic information transfer (PEXIT) algorithm. Also, the proposed new constructions of anti-jamming partially regular LDPC codes is based on follower jamming on the frequency-hopped spread spectrum (FHSS). For a channel environment, I suppose follower jamming with random dwell time and Rayleigh block fading environment with M-ary frequnecy shift keying (MFSK) modulation. For a coding perspective, an anti-jamming LDPC codes against follower jamming are introduced. In order to optimize the jamming environment, the partially regular structure and corresponding density evolution schemes are used. A series of simulations show that the proposed codes outperforms the 802.16e standard in the presence of follower noise jamming.이 논문에서는, i) 소실 채널에서 순환 부호의 새로운 이단 자기동형 군 복호기 , ii) 분산 저장 시스템을 위한 순환 부호 및 기존의 부분 접속 복구 부호(LRC)를 이용한 이진 혹은 삼진 부분 접속 복구 부호 설계법, 및 iii) 블록 간섭 환경을 위한 고부효율의 일반화된 근 프로토그래프(generalized root protograph, GRP) LDPC 부호 및 추적 재밍 환경을 위한 항재밍 부분 균일 (anti-jamming paritally regular, AJ-PR) LDPC 부호가 연구되었다. 첫번째로, 소실 채널에서 순환 부호의 새로운 이단 자기동형 군 복호기를 제안하였다. 최근 분산 저장 시스템 혹은 무선 센서 네트워크 등의 응용으로 인해 소실 채널에서의 오류 정정 부호 기법이 주목받고 있다. 그러나 많은 복호기 알고리즘은 높은 복호 복잡도 및 긴 지연으로 인해 실용적이지 못하다. 따라서 낮은 복호 복잡도 및 높은 성능을 보일 수 있는 순환 부호에서 이단 자기 동형 군 복호기가 제안되었다. 본 연구에서는 패리티 검사 행렬을 변형하고, 전처리 과정을 도입한 새로운 이단 자기동형 군 복호기를 제안한다. 제안한 복호기는 perfect 부호, BCH 부호 및 최대 거리 분리 (maximum distance separable, MDS) 부호에 대해서 분석되었다. 수치 분석을 통해, 제안된 복호 알고리즘은 기존의 자기 동형 군 복호기보다 낮은 복잡도를 보이며, 몇몇의 순환 부호 및 소실 채널에서 최대 우도 (maximal likelihood, ML)과 같은 수준의 성능임을 보인다. MDS 부호의 경우, 확장된 패리티검사 행렬 및 작은 크기의 행렬의 역연산을 활용하였을 경우의 성능을 분석한다. 두 번째로, 분산 저장 시스템을 위한 순환 부호 및 기존의 부분 접속 복구 부호 (LRC)를 이용한 이진 혹은 삼진 부분 접속 복구 부호 설계법을 제안하였다. 초기 연구로서, 순환 부호 및 연접을 활용한 이진 및 삼진 LRC 설계 기법이 연구되었다. 최소 해밍 거리가 4,5, 혹은 6인 제안된 이진 LRC 중 일부는 상한과 비교해 보았을 때 최적 설계임을 증명하였다. 또한, 비슷한 방법을 적용하여 좋은 파라미터의 삼진 LRC를 설계할 수 있었다. 그 외에 기존의 LRC를 활용하여 큰 해밍 거리의 새로운 LRC를 설계하는 방법을 제안하였다. 제안된 LRC는 분리된 복구 군 조건에서 최적이거나 최적에 가까운 값을 보였다. 마지막으로, GRP LDPC 부호는 Nakagami-$m$ 블록 페이딩 및 블록 간섭이 있는 두 상태의 이진 대칭 채널을 기반으로 한다. 이러한 채널 환경에서 GRP LDPC 부호는 하나의 블록 간섭이 발생했을 경우, 이론적 성능에 가까운 좋은 성능을 보여준다. 이러한 이론 값은 채널 문턱값이나 채널 outage 확률을 통해 검증할 수 있다. 제안된 설계에서는, 변형된 PEXIT 알고리즘을 활용하여 기초 행렬을 설계한다. 또한 AJ-PR LDPC 부호는 주파수 도약 환경에서 발생하는 추적 재밍이 있는 환경을 기반으로 한다. 채널 환경으로 MFSK 변복조 방식의 레일리 블록 페이딩 및 무작위한 지속 시간이 있는 재밍 환경을 가정한다. 이러한 재밍 환경으로 최적화하기 위해, 부분 균일 구조 및 해당되는 밀도 진화 (density evolution, DE) 기법이 활용된다. 여러 시뮬레이션 결과는 추적 재밍이 존재하는 환경에서 제안된 부호가 802.16e에 사용되었던 LDPC 부호보다 성능이 우수함을 보여준다.Contents Abstract Contents List of Tables List of Figures 1 INTRODUCTION 1.1 Background 1.2 Overview of Dissertation 1.3 Notations 2 Preliminaries 2.1 IED and AGD for Erasure Channel 2.1.1 Iterative Erasure Decoder 2.1.1 Automorphism Group Decoder 2.2. Binary Locally Repairable Codes for Distributed Storage System 2.2.1 Bounds and Optimalities of Binary LRCs 2.2.2 Existing Optimal Constructions of Binary LRCs 2.3 Channels with Block Interference and Jamming 2.3.1 Channels with Block Interference 2.3.2 Channels with Jamming with MFSK and FHSS Environment. 3 New Two-Stage Automorphism Group Decoders for Cyclic Codes in the Erasure Channel 3.1 Some Definitions 3.2 Modification of Parity Check Matrix and Two-Stage AGD 3.2.1 Modification of the Parity Check Matrix 3.2.2 A New Two-Stage AGD 3.2.3 Analysis of Modification Criteria for the Parity Check Matrix 3.2.4 Analysis of Decoding Complexity of TS-AGD 3.2.5 Numerical Analysis for Some Cyclic Codes 3.3 Construction of Parity Check Matrix and TS-AGD for Cyclic MDS Codes 3.3.1 Modification of Parity Check Matrix for Cyclic MDS Codes . 3.3.2 Proposed TS-AGD for Cyclic MDS Codes 3.3.3 Perfect Decoding by TS-AGD with Expanded Parity Check Matrix for Cyclic MDS Codes 3.3.4 TS-AGD with Submatrix Inversion for Cyclic MDS Codes . . 4 New Constructions of Binary and Ternary LRCs Using Cyclic Codes and Existing LRCs 4.1 Constructions of Binary LRCs Using Cyclic Codes 4.2 Constructions of Linear Ternary LRCs Using Cyclic Codes 4.3 Constructions of Binary LRCs with Disjoint Repair Groups Using Existing LRCs 4.4 New Constructions of Binary Linear LRCs with d ≥ 8 Using Existing LRCs 5 New Constructions of Generalized RP LDPC Codes for Block Interference and Partially Regular LDPC Codes for Follower Jamming 5.1 Generalized RP LDPC Codes for a Nonergodic BI 5.1.1 Minimum Blockwise Hamming Weight 5.1.2 Construction of GRP LDPC Codes 5.2 Asymptotic and Numerical Analyses of GRP LDPC Codes 5.2.1 Asymptotic Analysis of LDPC Codes 5.2.2 Numerical Analysis of Finite-Length LDPC Codes 5.3 Follower Noise Jamming with Fixed Scan Speed 5.4 Anti-Jamming Partially Regular LDPC Codes for Follower Noise Jamming 5.4.1 Simplified Channel Model and Corresponding Density Evolution 5.4.2 Construction of AJ-PR-LDPC Codes Based on DE 5.5 Numerical Analysis of AJ-PR LDPC Codes 6 Conclusion Abstract (In Korean)Docto

Low-Density Parity-Check Codes From Transversal Designs With Improved Stopping Set Distributions

This paper examines the construction of low-density parity-check (LDPC) codes from transversal designs based on sets of mutually orthogonal Latin squares (MOLS). By transferring the concept of configurations in combinatorial designs to the level of Latin squares, we thoroughly investigate the occurrence and avoidance of stopping sets for the arising codes. Stopping sets are known to determine the decoding performance over the binary erasure channel and should be avoided for small sizes. Based on large sets of simple-structured MOLS, we derive powerful constraints for the choice of suitable subsets, leading to improved stopping set distributions for the corresponding codes. We focus on LDPC codes with column weight 4, but the results are also applicable for the construction of codes with higher column weights. Finally, we show that a subclass of the presented codes has quasi-cyclic structure which allows low-complexity encoding.Comment: 11 pages; to appear in "IEEE Transactions on Communications