Binary and Ternary Quasi-perfect Codes with Small Dimensions

The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of infinite families of QP codes which includes all binary, ternary and quaternary codes known to is. We continue further with a list of sporadic examples of binary and ternary QP codes. Later we present the results of our investigation where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions up to 13 are classified.Comment: 4 page

Quasi-Perfect and Distance-Optimal Codes Sum-Rank Codes

Constructions of distance-optimal codes and quasi-perfect codes are challenging problems and have attracted many attentions. In this paper, we give the following three results. 1) If $\lambda|q^{sm}-1$ and $\lambda <\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+\epsilon)}}$, an infinite family of distance-optimal $q$-ary cyclic sum-rank codes with the block length $t=\frac{q^{sm}-1}{\lambda}$, the matrix size $s \times s$, the cardinality $q^{s^2t-s(2m+3)}$ and the minimum sum-rank distance four is constructed. 2) Block length $q^4-1$ and the matrix size $2 \times 2$ distance-optimal sum-rank codes with the minimum sum-rank distance four and the Singleton defect four are constructed. These sum-rank codes are close to the sphere packing bound , the Singleton-like bound and have much larger block length $q^4-1>>q-1$. 3) For given positive integers $m$ satisfying $2 \leq m$, an infinite family of quasi-perfect sum-rank codes with the matrix size $2 \times m$, and the minimum sum-rank distance three is also constructed. Quasi-perfect binary sum-rank codes with the minimum sum-rank distance four are also given. Almost MSRD $q$-ary codes with the block lengths up to $q^2$ are given. We show that more distance-optimal binary sum-rank codes can be obtained from the Plotkin sum.Comment: 19 pages, only quasi-perfect sum-rank codes were constructed. Almost MSRD codes with the block lengths up to $q^2$ were include

Deterministic Constructions of Binary Measurement Matrices from Finite Geometry

Deterministic constructions of measurement matrices in compressed sensing (CS) are considered in this paper. The constructions are inspired by the recent discovery of Dimakis, Smarandache and Vontobel which says that parity-check matrices of good low-density parity-check (LDPC) codes can be used as {provably} good measurement matrices for compressed sensing under $\ell_1$-minimization. The performance of the proposed binary measurement matrices is mainly theoretically analyzed with the help of the analyzing methods and results from (finite geometry) LDPC codes. Particularly, several lower bounds of the spark (i.e., the smallest number of columns that are linearly dependent, which totally characterizes the recovery performance of $\ell_0$-minimization) of general binary matrices and finite geometry matrices are obtained and they improve the previously known results in most cases. Simulation results show that the proposed matrices perform comparably to, sometimes even better than, the corresponding Gaussian random matrices. Moreover, the proposed matrices are sparse, binary, and most of them have cyclic or quasi-cyclic structure, which will make the hardware realization convenient and easy.Comment: 12 pages, 11 figure

50 Years of the Golomb--Welch Conjecture

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.Comment: 28 pages, 2 figure

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

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Evaporation-triggered microdroplet nucleation and the four life phases of an evaporating Ouzo drop

Evaporating liquid droplets are omnipresent in nature and technology, such as in inkjet printing, coating, deposition of materials, medical diagnostics, agriculture, food industry, cosmetics, or spills of liquids. While the evaporation of pure liquids, liquids with dispersed particles, or even liquid mixtures has intensively been studied over the last two decades, the evaporation of ternary mixtures of liquids with different volatilities and mutual solubilities has not yet been explored. Here we show that the evaporation of such ternary mixtures can trigger a phase transition and the nucleation of microdroplets of one of the components of the mixture. As model system we pick a sessile Ouzo droplet (as known from daily life - a transparent mixture of water, ethanol, and anise oil) and reveal and theoretically explain its four life phases: In phase I, the spherical cap-shaped droplet remains transparent, while the more volatile ethanol is evaporating, preferentially at the rim of the drop due to the singularity there. This leads to a local ethanol concentration reduction and correspondingly to oil droplet nucleation there. This is the beginning of phase II, in which oil microdroplets quickly nucleate in the whole drop, leading to its milky color which typifies the so-called 'Ouzo-effect'. Once all ethanol has evaporated, the drop, which now has a characteristic non-spherical-cap shape, has become clear again, with a water drop sitting on an oil-ring (phase III), finalizing the phase inversion. Finally, in phase IV, also all water has evaporated, leaving behind a tiny spherical cap-shaped oil drop.Comment: 40 pages, 12 figure