8 research outputs found
Bounds and Constructions of Singleton-Optimal Locally Repairable Codes with Small Localities
Constructions of optimal locally repairable codes (LRCs) achieving
Singleton-type bound have been exhaustively investigated in recent years. In
this paper, we consider new bounds and constructions of Singleton-optimal LRCs
with minmum distance , locality and minimum distance and
locality , respectively. Firstly, we establish equivalent connections
between the existence of these two families of LRCs and the existence of some
subsets of lines in the projective space with certain properties. Then, we
employ the line-point incidence matrix and Johnson bounds for constant weight
codes to derive new improved bounds on the code length, which are tighter than
known results. Finally, by using some techniques of finite field and finite
geometry, we give some new constructions of Singleton-optimal LRCs, which have
larger length than previous ones
Singleton-Optimal LRCs and Perfect LRCs via Cyclic and Constacyclic Codes
Locally repairable codes (LRCs) have emerged as an important coding scheme in
distributed storage systems (DSSs) with relatively low repair cost by accessing
fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs
have been widely investigated. Optimal LRCs via cyclic and constacyclic codes
provide significant benefit of elegant algebraic structure and efficient
encoding procedure. In this paper, we continue to consider the constructions of
optimal LRCs via cyclic and constacyclic codes with long code length.
Specifically, we first obtain two classes of -ary cyclic Singleton-optimal
-LRCs with length when and is
even, and length when and , respectively. To the best of our knowledge, this is the first
construction of -ary cyclic Singleton-optimal LRCs with length and
minimum distance . On the other hand, an LRC acheiving the
Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic
codes, we construct two new families of -ary perfect LRCs with length
, minimum distance and locality
Constructions of Binary Optimal Locally Repairable Codes via Intersection Subspaces
Locally repairable codes (LRCs), which can recover any symbol of a codeword
by reading only a small number of other symbols, have been widely used in
real-world distributed storage systems, such as Microsoft Azure Storage and
Ceph Storage Cluster. Since binary linear LRCs can significantly reduce coding
and decoding complexity, constructions of binary LRCs are of particular
interest. The aim of this paper is to construct dimensional optimal binary
locally repairable codes with disjoint local repair groups. We introduce how to
connect intersection subspaces with binary locally repairable codes and
construct dimensional optimal binary linear LRCs with locality () and minimum distance by employing intersection subspaces deduced
from the direct sum. This method will sufficiently increase the number of
possible repair groups of dimensional optimal LRCs, and thus efficiently
expanding the range of the construction parameters while keeping the largest
code rates compared with all known binary linear LRCs with minimum distance
and locality ().Comment: Accepted for publication in the SCIENCE CHINA Information Science
Bounds on the minimum distance of locally recoverable codes
We consider locally recoverable codes (LRCs) and aim to determine the
smallest possible length of a linear -code with
locality . For we exactly determine all values of and
for we exactly determine all values of . For the ternary
field we also state a few numerical results. As a general result we prove that
equals the Griesmer bound if the minimum Hamming distance is
sufficiently large and all other parameters are fixed.Comment: 23 pages, 3 table
์๋ก์ด ์์ค ์ฑ๋์ ์ํ ์๊ธฐ๋ํ ๊ตฐ ๋ณตํธ๊ธฐ ๋ฐ ๋ถ๋ถ ์ ์ ๋ณต๊ตฌ ๋ถํธ ๋ฐ ์ผ๋ฐํ๋ ๊ทผ ํ๋กํ ๊ทธ๋ํ LDPC ๋ถํธ์ ์ค๊ณ
ํ์๋
ผ๋ฌธ (๋ฐ์ฌ)-- ์์ธ๋ํ๊ต ๋ํ์ : ๊ณต๊ณผ๋ํ ์ ๊ธฐยท์ปดํจํฐ๊ณตํ๋ถ, 2019. 2. ๋
ธ์ข
์ .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- 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.์ด ๋
ผ๋ฌธ์์๋, i) ์์ค ์ฑ๋์์ ์ํ ๋ถํธ์ ์๋ก์ด ์ด๋จ ์๊ธฐ๋ํ ๊ตฐ ๋ณตํธ๊ธฐ , ii) ๋ถ์ฐ ์ ์ฅ ์์คํ
์ ์ํ ์ํ ๋ถํธ ๋ฐ ๊ธฐ์กด์ ๋ถ๋ถ ์ ์ ๋ณต๊ตฌ ๋ถํธ(LRC)๋ฅผ ์ด์ฉํ ์ด์ง ํน์ ์ผ์ง ๋ถ๋ถ ์ ์ ๋ณต๊ตฌ ๋ถํธ ์ค๊ณ๋ฒ, ๋ฐ iii) ๋ธ๋ก ๊ฐ์ญ ํ๊ฒฝ์ ์ํ ๊ณ ๋ถํจ์จ์ ์ผ๋ฐํ๋ ๊ทผ ํ๋กํ ๊ทธ๋ํ(generalized root protograph, GRP) LDPC ๋ถํธ ๋ฐ ์ถ์ ์ฌ๋ฐ ํ๊ฒฝ์ ์ํ ํญ์ฌ๋ฐ ๋ถ๋ถ ๊ท ์ผ (anti-jamming paritally regular, AJ-PR) LDPC ๋ถํธ๊ฐ ์ฐ๊ตฌ๋์๋ค.
์ฒซ๋ฒ์งธ๋ก, ์์ค ์ฑ๋์์ ์ํ ๋ถํธ์ ์๋ก์ด ์ด๋จ ์๊ธฐ๋ํ ๊ตฐ ๋ณตํธ๊ธฐ๋ฅผ ์ ์ํ์๋ค. ์ต๊ทผ ๋ถ์ฐ ์ ์ฅ ์์คํ
ํน์ ๋ฌด์ ์ผ์ ๋คํธ์ํฌ ๋ฑ์ ์์ฉ์ผ๋ก ์ธํด ์์ค ์ฑ๋์์์ ์ค๋ฅ ์ ์ ๋ถํธ ๊ธฐ๋ฒ์ด ์ฃผ๋ชฉ๋ฐ๊ณ ์๋ค. ๊ทธ๋ฌ๋ ๋ง์ ๋ณตํธ๊ธฐ ์๊ณ ๋ฆฌ์ฆ์ ๋์ ๋ณตํธ ๋ณต์ก๋ ๋ฐ ๊ธด ์ง์ฐ์ผ๋ก ์ธํด ์ค์ฉ์ ์ด์ง ๋ชปํ๋ค. ๋ฐ๋ผ์ ๋ฎ์ ๋ณตํธ ๋ณต์ก๋ ๋ฐ ๋์ ์ฑ๋ฅ์ ๋ณด์ผ ์ ์๋ ์ํ ๋ถํธ์์ ์ด๋จ ์๊ธฐ ๋ํ ๊ตฐ ๋ณตํธ๊ธฐ๊ฐ ์ ์๋์๋ค. ๋ณธ ์ฐ๊ตฌ์์๋ ํจ๋ฆฌํฐ ๊ฒ์ฌ ํ๋ ฌ์ ๋ณํํ๊ณ , ์ ์ฒ๋ฆฌ ๊ณผ์ ์ ๋์
ํ ์๋ก์ด ์ด๋จ ์๊ธฐ๋ํ ๊ตฐ ๋ณตํธ๊ธฐ๋ฅผ ์ ์ํ๋ค. ์ ์ํ ๋ณตํธ๊ธฐ๋ perfect ๋ถํธ, BCH ๋ถํธ ๋ฐ ์ต๋ ๊ฑฐ๋ฆฌ ๋ถ๋ฆฌ (maximum distance separable, MDS) ๋ถํธ์ ๋ํด์ ๋ถ์๋์๋ค. ์์น ๋ถ์์ ํตํด, ์ ์๋ ๋ณตํธ ์๊ณ ๋ฆฌ์ฆ์ ๊ธฐ์กด์ ์๊ธฐ ๋ํ ๊ตฐ ๋ณตํธ๊ธฐ๋ณด๋ค ๋ฎ์ ๋ณต์ก๋๋ฅผ ๋ณด์ด๋ฉฐ, ๋ช๋ช์ ์ํ ๋ถํธ ๋ฐ ์์ค ์ฑ๋์์ ์ต๋ ์ฐ๋ (maximal likelihood, ML)๊ณผ ๊ฐ์ ์์ค์ ์ฑ๋ฅ์์ ๋ณด์ธ๋ค. MDS ๋ถํธ์ ๊ฒฝ์ฐ, ํ์ฅ๋ ํจ๋ฆฌํฐ๊ฒ์ฌ ํ๋ ฌ ๋ฐ ์์ ํฌ๊ธฐ์ ํ๋ ฌ์ ์ญ์ฐ์ฐ์ ํ์ฉํ์์ ๊ฒฝ์ฐ์ ์ฑ๋ฅ์ ๋ถ์ํ๋ค.
๋ ๋ฒ์งธ๋ก, ๋ถ์ฐ ์ ์ฅ ์์คํ
์ ์ํ ์ํ ๋ถํธ ๋ฐ ๊ธฐ์กด์ ๋ถ๋ถ ์ ์ ๋ณต๊ตฌ ๋ถํธ (LRC)๋ฅผ ์ด์ฉํ ์ด์ง ํน์ ์ผ์ง ๋ถ๋ถ ์ ์ ๋ณต๊ตฌ ๋ถํธ ์ค๊ณ๋ฒ์ ์ ์ํ์๋ค. ์ด๊ธฐ ์ฐ๊ตฌ๋ก์, ์ํ ๋ถํธ ๋ฐ ์ฐ์ ์ ํ์ฉํ ์ด์ง ๋ฐ ์ผ์ง LRC ์ค๊ณ ๊ธฐ๋ฒ์ด ์ฐ๊ตฌ๋์๋ค. ์ต์ ํด๋ฐ ๊ฑฐ๋ฆฌ๊ฐ 4,5, ํน์ 6์ธ ์ ์๋ ์ด์ง LRC ์ค ์ผ๋ถ๋ ์ํ๊ณผ ๋น๊ตํด ๋ณด์์ ๋ ์ต์ ์ค๊ณ์์ ์ฆ๋ช
ํ์๋ค. ๋ํ, ๋น์ทํ ๋ฐฉ๋ฒ์ ์ ์ฉํ์ฌ ์ข์ ํ๋ผ๋ฏธํฐ์ ์ผ์ง LRC๋ฅผ ์ค๊ณํ ์ ์์๋ค. ๊ทธ ์ธ์ ๊ธฐ์กด์ LRC๋ฅผ ํ์ฉํ์ฌ ํฐ ํด๋ฐ ๊ฑฐ๋ฆฌ์ ์๋ก์ด LRC๋ฅผ ์ค๊ณํ๋ ๋ฐฉ๋ฒ์ ์ ์ํ์๋ค. ์ ์๋ LRC๋ ๋ถ๋ฆฌ๋ ๋ณต๊ตฌ ๊ตฐ ์กฐ๊ฑด์์ ์ต์ ์ด๊ฑฐ๋ ์ต์ ์ ๊ฐ๊น์ด ๊ฐ์ ๋ณด์๋ค.
๋ง์ง๋ง์ผ๋ก, GRP LDPC ๋ถํธ๋ Nakagami- ๋ธ๋ก ํ์ด๋ฉ ๋ฐ ๋ธ๋ก ๊ฐ์ญ์ด ์๋ ๋ ์ํ์ ์ด์ง ๋์นญ ์ฑ๋์ ๊ธฐ๋ฐ์ผ๋ก ํ๋ค. ์ด๋ฌํ ์ฑ๋ ํ๊ฒฝ์์ GRP LDPC ๋ถํธ๋ ํ๋์ ๋ธ๋ก ๊ฐ์ญ์ด ๋ฐ์ํ์ ๊ฒฝ์ฐ, ์ด๋ก ์ ์ฑ๋ฅ์ ๊ฐ๊น์ด ์ข์ ์ฑ๋ฅ์ ๋ณด์ฌ์ค๋ค. ์ด๋ฌํ ์ด๋ก ๊ฐ์ ์ฑ๋ ๋ฌธํฑ๊ฐ์ด๋ ์ฑ๋ outage ํ๋ฅ ์ ํตํด ๊ฒ์ฆํ ์ ์๋ค. ์ ์๋ ์ค๊ณ์์๋, ๋ณํ๋ PEXIT ์๊ณ ๋ฆฌ์ฆ์ ํ์ฉํ์ฌ ๊ธฐ์ด ํ๋ ฌ์ ์ค๊ณํ๋ค. ๋ํ AJ-PR LDPC ๋ถํธ๋ ์ฃผํ์ ๋์ฝ ํ๊ฒฝ์์ ๋ฐ์ํ๋ ์ถ์ ์ฌ๋ฐ์ด ์๋ ํ๊ฒฝ์ ๊ธฐ๋ฐ์ผ๋ก ํ๋ค. ์ฑ๋ ํ๊ฒฝ์ผ๋ก MFSK ๋ณ๋ณต์กฐ ๋ฐฉ์์ ๋ ์ผ๋ฆฌ ๋ธ๋ก ํ์ด๋ฉ ๋ฐ ๋ฌด์์ํ ์ง์ ์๊ฐ์ด ์๋ ์ฌ๋ฐ ํ๊ฒฝ์ ๊ฐ์ ํ๋ค. ์ด๋ฌํ ์ฌ๋ฐ ํ๊ฒฝ์ผ๋ก ์ต์ ํํ๊ธฐ ์ํด, ๋ถ๋ถ ๊ท ์ผ ๊ตฌ์กฐ ๋ฐ ํด๋น๋๋ ๋ฐ๋ ์งํ (density evolution, DE) ๊ธฐ๋ฒ์ด ํ์ฉ๋๋ค. ์ฌ๋ฌ ์๋ฎฌ๋ ์ด์
๊ฒฐ๊ณผ๋ ์ถ์ ์ฌ๋ฐ์ด ์กด์ฌํ๋ ํ๊ฒฝ์์ ์ ์๋ ๋ถํธ๊ฐ 802.16e์ ์ฌ์ฉ๋์๋ LDPC ๋ถํธ๋ณด๋ค ์ฑ๋ฅ์ด ์ฐ์ํจ์ ๋ณด์ฌ์ค๋ค.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 โฅ 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