Spatially Coupled LDPC Codes Constructed from Protographs

In this paper, we construct protograph-based spatially coupled low-density parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L, we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary L. We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large L, the BP thresholds on both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (AWGNC) saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output (MBS) channels with low-complexity BP decoding.Comment: Submitted to the IEEE Transactions on Information Theor

Windowed Decoding of Protograph-based LDPC Convolutional Codes over Erasure Channels

We consider a windowed decoding scheme for LDPC convolutional codes that is based on the belief-propagation (BP) algorithm. We discuss the advantages of this decoding scheme and identify certain characteristics of LDPC convolutional code ensembles that exhibit good performance with the windowed decoder. We will consider the performance of these ensembles and codes over erasure channels with and without memory. We show that the structure of LDPC convolutional code ensembles is suitable to obtain performance close to the theoretical limits over the memoryless erasure channel, both for the BP decoder and windowed decoding. However, the same structure imposes limitations on the performance over erasure channels with memory.Comment: 18 pages, 9 figures, accepted for publication in the IEEE Transactions on Information Theor

New Protograph-Based Construction of GLDPC Codes for Binary Erasure Channel and LDPC Codes for Block Fading Channel

학위논문(박사) -- 서울대학교대학원 : 공과대학 전기·정보공학부, 2022.2. 노종선 교수님.이 학위 논문에서는 다음 두 가지의 연구가 이루어졌다: i) 이진 소실 채널에서 새로운 구조의 프로토그래프 기반 generalized low-density parity-check (GLDPC) 부호의 설계 방법 ii) 블록 페이딩 채널을 위한 프로토그래프 기반의 LDPC 부호 설계. 첫 번째로, 이진 소실 채널에서 새롭게 제안된 부분적 도핑 기법을 이용한 프로토그래프 기반의 GLDPC 부호가 제안되었다. 기존의 프로토그래프 기반의 GLDPC 부호의 경우 프로토그래프 영역에서 single parity-check (SPC) 노드를 generalized constraint (GC) 노드로 치환(도핑)하는 형태로 부호가 설계되어 여러 변수 노드 걸쳐 GC 노드가 연결되는 형태를 가진다. 반면, 제안된 부분적 도핑 기법은 한 개의 변수 노드에 GC 노드를 연결하도록 만들 수 있다. 바꿔 말하면, 제안된 부분적 도핑 기법은 더 세밀한 도핑이 가능해서 결과적으로 부호 설계에 있어 높은 자유도를 가지고 더 세련된 부호 최적화가 가능하다. 본 학위 논문에서는 부분적 도핑과 PEXIT 분석을 이용하여 partially doped GLDPC (PD-GLDPC) 부호를 설계하고 최적화 하였다. 더불어, PD-GLDPC 부호의 일반적 최소 거리를 가지는 조건을 제시하였고 이를 이 론적으로 증명하였다. 결과적으로, 제안된 PD-GLDPC 부호는 현존하는 GLDPC 부호의 성능보다 유의미하게 워터플 성능이 좋았고 동시에 오류 마루가 없었다. 마지막으로, 최적화된 PD-GLDPC 부호는 현존하는 최신 블록 LDPC 부호들에 근접한 성능을 가짐을 보여주었다. 두 번째로, 블록 페이딩 (BF) 채널에서 resolvable block design (RBD)를 이용한 프로토그래프 LDPC 부호 설계가 이루어졌다. 제안된 부호의 성능을 확인하기 위한 비트 오류율의 상한을 감마 진화라는 제안된 기법을 이용해 유도하였다. 또한, 시뮬레이션을 통해 유도된 오류율 상한과 부호의 프레임 오류율이 높은 SNR 영역에서 채널 outage 확률에 근접함을 알 수 있다.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) 78박

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 José Murillo Fuentes.- Secretario: Matilde Pilar Sánchez Fernández.- Vocal: Javier Valls Coquilla

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

Spatially Coupled Turbo-Like Codes

The focus of this thesis is on proposing and analyzing a powerful class of codes on graphs---with trellis constraints---that can simultaneously approach capacity and achieve very low error floor. In particular, we propose the concept of spatial coupling for turbo-like code (SC-TC) ensembles and investigate the impact of coupling on the performance of these codes. The main elements of this study can be summarized by the following four major topics. First, we considered the spatial coupling of parallel concatenated codes (PCCs), serially concatenated codes (SCCs), and hybrid concatenated codes (HCCs).We also proposed two extensions of braided convolutional codes (BCCs) to higher coupling memories. Second, we investigated the impact of coupling on the asymptotic behavior of the proposed ensembles in term of the decoding thresholds. For that, we derived the exact density evolution (DE) equations of the proposed SC-TC ensembles over the binary erasure channel. Using the DE equations, we found the thresholds of the coupled and uncoupled ensembles under belief propagation (BP) decoding for a wide range of rates. We also computed the maximum a-posteriori (MAP) thresholds of the underlying uncoupled ensembles. Our numerical results confirm that TCs have excellent MAP thresholds, and for a large enough coupling memory, the BP threshold of an SC-TC ensemble improves to the MAP threshold of the underlying TC ensemble. This phenomenon is called threshold saturation and we proved its occurrence for SC-TCs by use of a proof technique based on the potential function of the ensembles.Third, we investigated and discussed the performance of SC-TCs in the finite length regime. We proved that under certain conditions the minimum distance of an SC-TCs is either larger or equal to that of its underlying uncoupled ensemble. Based on this fact, we performed a weight enumerator (WE) analysis for the underlying uncoupled ensembles to investigate the error floor performance of the SC-TC ensembles. We computed bounds on the error rate performance and minimum distance of the TC ensembles. These bounds indicate very low error floor for SCC, HCC, and BCC ensembles, and show that for HCC, and BCC ensembles, the minimum distance grows linearly with the input block length.The results from the DE and WE analysis demonstrate that the performance of TCs benefits from spatial coupling in both waterfall and error floor regions. While uncoupled TC ensembles with close-to-capacity performance exhibit a high error floor, our results show that SC-TCs can simultaneously approach capacity and achieve very low error floor.Fourth, we proposed a unified ensemble of TCs that includes all the considered TC classes. We showed that for each of the original classes of TCs, it is possible to find an equivalent ensemble by proper selection of the design parameters in the unified ensemble. This unified ensemble not only helps us to understand the connections and trade-offs between the TC ensembles but also can be considered as a bridge between TCs and generalized low-density parity check codes

Multiplicatively Repeated Non-Binary LDPC Codes

We propose non-binary LDPC codes concatenated with multiplicative repetition codes. By multiplicatively repeating the (2,3)-regular non-binary LDPC mother code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9, 1/12,... Surprisingly, such simple low-rate non-binary LDPC codes outperform the best low-rate binary LDPC codes so far. Moreover, we propose the decoding algorithm for the proposed codes, which can be decoded with almost the same computational complexity as that of the mother code.Comment: To appear in IEEE Transactions on Information Theor