Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes

We present $O(m^3)$ algorithms for specifying the support of minimum-weight words of extended binary BCH codes of length $n=2^m$ and designed distance $d(m,s,i):=2^{m-1-s}-2^{m-1-i-s}$ for some values of $m,i,s$, where $m$ may grow to infinity. The support is specified as the sum of two sets: a set of $2^{2i-1}-2^{i-1}$ elements, and a subspace of dimension $m-2i-s$, specified by a basis. In some detail, for designed distance $6\cdot 2^j$, we have a deterministic algorithm for even $m\geq 4$, and a probabilistic algorithm with success probability $1-O(2^{-m})$ for odd $m>4$. For designed distance $28\cdot 2^j$, we have a probabilistic algorithm with success probability $\geq 1/3-O(2^{-m/2})$ for even $m\geq 6$. Finally, for designed distance $120\cdot 2^j$, we have a deterministic algorithm for $m\geq 8$ divisible by $4$. We also present a construction via Gold functions when $2i|m$. Our construction builds on results of Kasami and Lin (IEEE T-IT, 1972), who proved that for extended binary BCH codes of designed distance $d(m,s,i)$, the minimum distance equals the designed distance. Their proof makes use of a non-constructive result of Berlekamp (Inform. Contrl., 1970), and a constructive ``down-conversion theorem'' that converts some words in BCH codes to lower-weight words in BCH codes of lower designed distance. Our main contribution is in replacing the non-constructive argument of Berlekamp by a low-complexity algorithm. In one aspect, we extends the results of Grigorescu and Kaufman (IEEE T-IT, 2012), who presented explicit minimum-weight words for designed distance $6$ (and hence also for designed distance $6\cdot 2^j$, by a well-known ``up-conversion theorem''), as we cover more cases of the minimum distance. However, the minimum-weight words we construct are not affine generators for designed distance $>6$

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

Generalized discrete Fourier transform with non-linear phase : theory and design

Constant modulus transforms like discrete Fourier transform (DFT), Walsh transform, and Gold codes have been successfully used over several decades in various engineering applications, including discrete multi-tone (DMT), orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA) communications systems. Among these popular transforms, DFT is a linear phase transform and widely used in multicarrier communications due to its performance and fast algorithms. In this thesis, a theoretical framework for Generalized DFT (GDFT) with nonlinear phase exploiting the phase space is developed. It is shown that GDFT offers sizable correlation improvements over DFT, Walsh, and Gold codes. Brute force search algorithm is employed to obtain orthogonal GDFT code sets with improved correlations. Design examples and simulation results on several channel types presented in the thesis show that the proposed GDFT codes, with better auto and cross-correlation properties than DFT, lead to better bit-error-rate performance in all multi-carrier and multi-user communications scenarios investigated. It is also highlighted how known constant modulus code families such as Walsh, Walsh-like and other codes are special solutions of the GDFT framework. In addition to theoretical framework, practical design methods with computationally efficient implementations of GDFT as enhancements to DFT are presented in the thesis. The main advantage of the proposed method is its ability to design a wide selection of constant modulus orthogonal code sets based on the desired performance metrics mimicking the engineering .specs of interest. Orthogonal Frequency Division Multiplexing (OFDM) is a leading candidate to be adopted for high speed 4G wireless communications standards due to its high spectral efficiency, strong resistance to multipath fading and ease of implementation with Fast Fourier Transform (FFT) algorithms. However, the main disadvantage of an OFDM based communications technique is of its high PAPR at the RF stage of a transmitter. PAPR dominates the power (battery) efficiency of the radio transceiver. Among the PAPR reduction methods proposed in the literature, Selected Mapping (SLM) method has been successfully used in OFDM communications. In this thesis, an SLM method employing GDFT with closed form phase functions rather than fixed DFT for PAPR reduction is introduced. The performance improvements of GDFT based SLM PAPR reduction for various OFDM communications scenarios including the WiMAX standard based system are evaluated by simulations. Moreover, an efficient implementation of GDFT based SLM method reducing computational cost of multiple transform operations is forwarded. Performance simulation results show that power efficiency of non-linear RF amplifier in an OFDM system employing proposed method significantly improved

Equivalence Theorems and the Local-Global Property

In this thesis we revisit some classical results about the MacWilliams equivalence theorems for codes over fields and rings. These theorems deal with the question whether, for a given weight function, weight-preserving isomorphisms between codes can be described explicitly. We will show that a condition, which was already known to be sufficient for the MacWilliams equivalence theorem, is also necessary. Furthermore we will study a local-global property that naturally generalizes the MacWilliams equivalence theorems. Making use of F-partitions, we will prove that for various subgroups of the group of invertible matrices the local-global extension principle is valid

Orthogonal transmultiplexers : extensions to digital subscriber line (DSL) communications

An orthogonal transmultiplexer which unifies multirate filter bank theory and communications theory is investigated in this dissertation. Various extensions of the orthogonal transmultiplexer techniques have been made for digital subscriber line communication applications. It is shown that the theoretical performance bounds of single carrier modulation based transceivers and multicarrier modulation based transceivers are the same under the same operational conditions. Single carrier based transceiver systems such as Quadrature Amplitude Modulation (QAM) and Carrierless Amplitude and Phase (CAP) modulation scheme, multicarrier based transceiver systems such as Orthogonal Frequency Division Multiplexing (OFDM) or Discrete Multi Tone (DMT) and Discrete Subband (Wavelet) Multicarrier based transceiver (DSBMT) techniques are considered in this investigation. The performance of DMT and DSBMT based transceiver systems for a narrow band interference and their robustness are also investigated. It is shown that the performance of a DMT based transceiver system is quite sensitive to the location and strength of a single tone (narrow band) interference. The performance sensitivity is highlighted in this work. It is shown that an adaptive interference exciser can alleviate the sensitivity problem of a DMT based system. The improved spectral properties of DSBMT technique reduces the performance sensitivity for variations of a narrow band interference. It is shown that DSBMT technique outperforms DMT and has a more robust performance than the latter. The superior performance robustness is shown in this work. Optimal orthogonal basis design using cosine modulated multirate filter bank is discussed. An adaptive linear combiner at the output of analysis filter bank is implemented to eliminate the intersymbol and interchannel interferences. It is shown that DSBMT is the most suitable technique for a narrow band interference environment. A blind channel identification and optimal MMSE based equalizer employing a nonmaximally decimated filter bank precoder / postequalizer structure is proposed. The performance of blind channel identification scheme is shown not to be sensitive to the characteristics of unknown channel. The performance of the proposed optimal MMSE based equalizer is shown to be superior to the zero-forcing equalizer

Mathematical aspects of the design and security of block ciphers

Block ciphers constitute a major part of modern symmetric cryptography. A mathematical analysis is necessary to ensure the security of the cipher. In this thesis, I develop several new contributions for the analysis of block ciphers. I determine cryptographic properties of several special cryptographically interesting mappings like almost perfect nonlinear functions. I also give some new results both on the resistance of functions against differential-linear attacks as well as on the efficiency of implementation of certain block ciphers

Structure-based classification and ontology in chemistry

<p>Abstract</p> <p>Background</p> <p>Recent years have seen an explosion in the availability of data in the chemistry domain. With this information explosion, however, retrieving <it>relevant </it>results from the available information, and <it>organising </it>those results, become even harder problems. Computational processing is essential to filter and organise the available resources so as to better facilitate the work of scientists. Ontologies encode expert domain knowledge in a hierarchically organised machine-processable format. One such ontology for the chemical domain is ChEBI. ChEBI provides a classification of chemicals based on their structural features and a role or activity-based classification. An example of a structure-based class is 'pentacyclic compound' (compounds containing five-ring structures), while an example of a role-based class is 'analgesic', since many different chemicals can act as analgesics without sharing structural features. Structure-based classification in chemistry exploits elegant regularities and symmetries in the underlying chemical domain. As yet, there has been neither a systematic analysis of the types of structural classification in use in chemistry nor a comparison to the capabilities of available technologies.</p> <p>Results</p> <p>We analyze the different categories of structural classes in chemistry, presenting a list of patterns for features found in class definitions. We compare these patterns of class definition to tools which allow for automation of hierarchy construction within cheminformatics and within logic-based ontology technology, going into detail in the latter case with respect to the expressive capabilities of the Web Ontology Language and recent extensions for modelling structured objects. Finally we discuss the relationships and interactions between cheminformatics approaches and logic-based approaches.</p> <p>Conclusion</p> <p>Systems that perform intelligent reasoning tasks on chemistry data require a diverse set of underlying computational utilities including algorithmic, statistical and logic-based tools. For the task of automatic structure-based classification of chemical entities, essential to managing the vast swathes of chemical data being brought online, systems which are capable of hybrid reasoning combining several different approaches are crucial. We provide a thorough review of the available tools and methodologies, and identify areas of open research.</p