Two Optimal One-Error-Correcting Codes of Length 13 That Are Not Doubly Shortened Perfect Codes

The doubly shortened perfect codes of length 13 are classified utilizing the classification of perfect codes in [P.R.J. \"Osterg{\aa}rd and O. Pottonen, The perfect binary one-error-correcting codes of length 15: Part I - Classification, IEEE Trans. Inform. Theory, to appear]; there are 117821 such (13,512,3) codes. By applying a switching operation to those codes, two more (13,512,3) codes are obtained, which are then not doubly shortened perfect codes.Comment: v2: a correction concerning shortened codes of length 1

On Optimal Binary One-Error-Correcting Codes of Lengths 2m−42^m-4 and 2m−32^m-3

Best and Brouwer [Discrete Math. 17 (1977), 235-245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length $2^m-4$ and $2^m-3$, respectively) are optimal. Properties of such codes are here studied, determining among other things parameters of certain subcodes. A utilization of these properties makes a computer-aided classification of the optimal binary one-error-correcting codes of lengths 12 and 13 possible; there are 237610 and 117823 such codes, respectively (with 27375 and 17513 inequivalent extensions). This completes the classification of optimal binary one-error-correcting codes for all lengths up to 15. Some properties of the classified codes are further investigated. Finally, it is proved that for any $m \geq 4$, there are optimal binary one-error-correcting codes of length $2^m-4$ and $2^m-3$ that cannot be lengthened to perfect codes of length $2^m-1$.Comment: Accepted for publication in IEEE Transactions on Information Theory. Data available at http://www.iki.fi/opottone/code

On the binary codes with parameters of triply-shortened 1-perfect codes

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the $n$-cube into six cells. An arbitrary binary $(n=2^m-4, 2^{n-m}, 3)$ code $D$, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) from six cells. As a corollary, the codes $C$ and $D$ are completely semiregular; i.e., the weight distribution of such a code depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if $D$ is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable. Keywords: 1-perfect code; triply-shortened 1-perfect code; equitable partition; perfect coloring; weight distribution; distance distributionComment: 12 page

On qq-ary shortened-11-perfect-like codes

We study codes with parameters of $q$-ary shortened Hamming codes, i.e., $(n=(q^m-q)/(q-1), q^{n-m}, 3)_q$. At first, we prove the fact mentioned in [A.E.Brouwer et al. Bounds on mixed binary/ternary codes. IEEE Trans. Inf. Theory 44 (1998) 140-161] that such codes are optimal, generalizing it to a bound for multifold packings of radius-$1$ balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal $q$-fold packing with minimum distance $2$. At second, we show the existence of $4$-ary codes with parameters of shortened $1$-perfect codes that cannot be obtained by shortening a $1$-perfect code. Keywords: Hamming graph; multifold packings; multiple coverings; perfect codes

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