276 research outputs found

    Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes

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    We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on Information Theor

    Efficient Linear Programming Decoding of HDPC Codes

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    We propose several improvements for Linear Programming (LP) decoding algorithms for High Density Parity Check (HDPC) codes. First, we use the automorphism groups of a code to create parity check matrix diversity and to generate valid cuts from redundant parity checks. Second, we propose an efficient mixed integer decoder utilizing the branch and bound method. We further enhance the proposed decoders by removing inactive constraints and by adapting the parity check matrix prior to decoding according to the channel observations. Based on simulation results the proposed decoders achieve near-ML performance with reasonable complexity.Comment: Submitted to the IEEE Transactions on Communications, November 200

    Efficient Maximum-Likelihood Decoding of Linear Block Codes on Binary Memoryless Channels

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    In this work, we consider efficient maximum-likelihood decoding of linear block codes for small-to-moderate block lengths. The presented approach is a branch-and-bound algorithm using the cutting-plane approach of Zhang and Siegel (IEEE Trans. Inf. Theory, 2012) for obtaining lower bounds. We have compared our proposed algorithm to the state-of-the-art commercial integer program solver CPLEX, and for all considered codes our approach is faster for both low and high signal-to-noise ratios. For instance, for the benchmark (155,64) Tanner code our algorithm is more than 11 times as fast as CPLEX for an SNR of 1.0 dB on the additive white Gaussian noise channel. By a small modification, our algorithm can be used to calculate the minimum distance, which we have again verified to be much faster than using the CPLEX solver.Comment: Submitted to 2014 International Symposium on Information Theory. 5 Pages. Accepte

    Deterministic Construction of Binary, Bipolar and Ternary Compressed Sensing Matrices

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    In this paper we establish the connection between the Orthogonal Optical Codes (OOC) and binary compressed sensing matrices. We also introduce deterministic bipolar mร—nm\times n RIP fulfilling ยฑ1\pm 1 matrices of order kk such that mโ‰คO(k(logโก2n)logโก2klnโกlogโก2k)m\leq\mathcal{O}\big(k (\log_2 n)^{\frac{\log_2 k}{\ln \log_2 k}}\big). The columns of these matrices are binary BCH code vectors where the zeros are replaced by -1. Since the RIP is established by means of coherence, the simple greedy algorithms such as Matching Pursuit are able to recover the sparse solution from the noiseless samples. Due to the cyclic property of the BCH codes, we show that the FFT algorithm can be employed in the reconstruction methods to considerably reduce the computational complexity. In addition, we combine the binary and bipolar matrices to form ternary sensing matrices ({0,1,โˆ’1}\{0,1,-1\} elements) that satisfy the RIP condition.Comment: The paper is accepted for publication in IEEE Transaction on Information Theor

    ์ƒˆ๋กœ์šด ์†Œ์‹ค ์ฑ„๋„์„ ์œ„ํ•œ ์ž๊ธฐ๋™ํ˜• ๊ตฐ ๋ณตํ˜ธ๊ธฐ ๋ฐ ๋ถ€๋ถ„ ์ ‘์† ๋ณต๊ตฌ ๋ถ€ํ˜ธ ๋ฐ ์ผ๋ฐ˜ํ™”๋œ ๊ทผ ํ”„๋กœํ† ๊ทธ๋ž˜ํ”„ LDPC ๋ถ€ํ˜ธ์˜ ์„ค๊ณ„

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    ํ•™์œ„๋…ผ๋ฌธ (๋ฐ•์‚ฌ)-- ์„œ์šธ๋Œ€ํ•™๊ต ๋Œ€ํ•™์› : ๊ณต๊ณผ๋Œ€ํ•™ ์ „๊ธฐยท์ปดํ“จํ„ฐ๊ณตํ•™๋ถ€, 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-mm 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 ๋…ผ๋ฌธ์—์„œ๋Š”, i) ์†Œ์‹ค ์ฑ„๋„์—์„œ ์ˆœํ™˜ ๋ถ€ํ˜ธ์˜ ์ƒˆ๋กœ์šด ์ด๋‹จ ์ž๊ธฐ๋™ํ˜• ๊ตฐ ๋ณตํ˜ธ๊ธฐ , ii) ๋ถ„์‚ฐ ์ €์žฅ ์‹œ์Šคํ…œ์„ ์œ„ํ•œ ์ˆœํ™˜ ๋ถ€ํ˜ธ ๋ฐ ๊ธฐ์กด์˜ ๋ถ€๋ถ„ ์ ‘์† ๋ณต๊ตฌ ๋ถ€ํ˜ธ(LRC)๋ฅผ ์ด์šฉํ•œ ์ด์ง„ ํ˜น์€ ์‚ผ์ง„ ๋ถ€๋ถ„ ์ ‘์† ๋ณต๊ตฌ ๋ถ€ํ˜ธ ์„ค๊ณ„๋ฒ•, ๋ฐ iii) ๋ธ”๋ก ๊ฐ„์„ญ ํ™˜๊ฒฝ์„ ์œ„ํ•œ ๊ณ ๋ถ€ํšจ์œจ์˜ ์ผ๋ฐ˜ํ™”๋œ ๊ทผ ํ”„๋กœํ† ๊ทธ๋ž˜ํ”„(generalized root protograph, GRP) LDPC ๋ถ€ํ˜ธ ๋ฐ ์ถ”์  ์žฌ๋ฐ ํ™˜๊ฒฝ์„ ์œ„ํ•œ ํ•ญ์žฌ๋ฐ ๋ถ€๋ถ„ ๊ท ์ผ (anti-jamming paritally regular, AJ-PR) LDPC ๋ถ€ํ˜ธ๊ฐ€ ์—ฐ๊ตฌ๋˜์—ˆ๋‹ค. ์ฒซ๋ฒˆ์งธ๋กœ, ์†Œ์‹ค ์ฑ„๋„์—์„œ ์ˆœํ™˜ ๋ถ€ํ˜ธ์˜ ์ƒˆ๋กœ์šด ์ด๋‹จ ์ž๊ธฐ๋™ํ˜• ๊ตฐ ๋ณตํ˜ธ๊ธฐ๋ฅผ ์ œ์•ˆํ•˜์˜€๋‹ค. ์ตœ๊ทผ ๋ถ„์‚ฐ ์ €์žฅ ์‹œ์Šคํ…œ ํ˜น์€ ๋ฌด์„  ์„ผ์„œ ๋„คํŠธ์›Œํฌ ๋“ฑ์˜ ์‘์šฉ์œผ๋กœ ์ธํ•ด ์†Œ์‹ค ์ฑ„๋„์—์„œ์˜ ์˜ค๋ฅ˜ ์ •์ • ๋ถ€ํ˜ธ ๊ธฐ๋ฒ•์ด ์ฃผ๋ชฉ๋ฐ›๊ณ  ์žˆ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ๋งŽ์€ ๋ณตํ˜ธ๊ธฐ ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ ๋†’์€ ๋ณตํ˜ธ ๋ณต์žก๋„ ๋ฐ ๊ธด ์ง€์—ฐ์œผ๋กœ ์ธํ•ด ์‹ค์šฉ์ ์ด์ง€ ๋ชปํ•˜๋‹ค. ๋”ฐ๋ผ์„œ ๋‚ฎ์€ ๋ณตํ˜ธ ๋ณต์žก๋„ ๋ฐ ๋†’์€ ์„ฑ๋Šฅ์„ ๋ณด์ผ ์ˆ˜ ์žˆ๋Š” ์ˆœํ™˜ ๋ถ€ํ˜ธ์—์„œ ์ด๋‹จ ์ž๊ธฐ ๋™ํ˜• ๊ตฐ ๋ณตํ˜ธ๊ธฐ๊ฐ€ ์ œ์•ˆ๋˜์—ˆ๋‹ค. ๋ณธ ์—ฐ๊ตฌ์—์„œ๋Š” ํŒจ๋ฆฌํ‹ฐ ๊ฒ€์‚ฌ ํ–‰๋ ฌ์„ ๋ณ€ํ˜•ํ•˜๊ณ , ์ „์ฒ˜๋ฆฌ ๊ณผ์ •์„ ๋„์ž…ํ•œ ์ƒˆ๋กœ์šด ์ด๋‹จ ์ž๊ธฐ๋™ํ˜• ๊ตฐ ๋ณตํ˜ธ๊ธฐ๋ฅผ ์ œ์•ˆํ•œ๋‹ค. ์ œ์•ˆํ•œ ๋ณตํ˜ธ๊ธฐ๋Š” perfect ๋ถ€ํ˜ธ, BCH ๋ถ€ํ˜ธ ๋ฐ ์ตœ๋Œ€ ๊ฑฐ๋ฆฌ ๋ถ„๋ฆฌ (maximum distance separable, MDS) ๋ถ€ํ˜ธ์— ๋Œ€ํ•ด์„œ ๋ถ„์„๋˜์—ˆ๋‹ค. ์ˆ˜์น˜ ๋ถ„์„์„ ํ†ตํ•ด, ์ œ์•ˆ๋œ ๋ณตํ˜ธ ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ ๊ธฐ์กด์˜ ์ž๊ธฐ ๋™ํ˜• ๊ตฐ ๋ณตํ˜ธ๊ธฐ๋ณด๋‹ค ๋‚ฎ์€ ๋ณต์žก๋„๋ฅผ ๋ณด์ด๋ฉฐ, ๋ช‡๋ช‡์˜ ์ˆœํ™˜ ๋ถ€ํ˜ธ ๋ฐ ์†Œ์‹ค ์ฑ„๋„์—์„œ ์ตœ๋Œ€ ์šฐ๋„ (maximal likelihood, ML)๊ณผ ๊ฐ™์€ ์ˆ˜์ค€์˜ ์„ฑ๋Šฅ์ž„์„ ๋ณด์ธ๋‹ค. MDS ๋ถ€ํ˜ธ์˜ ๊ฒฝ์šฐ, ํ™•์žฅ๋œ ํŒจ๋ฆฌํ‹ฐ๊ฒ€์‚ฌ ํ–‰๋ ฌ ๋ฐ ์ž‘์€ ํฌ๊ธฐ์˜ ํ–‰๋ ฌ์˜ ์—ญ์—ฐ์‚ฐ์„ ํ™œ์šฉํ•˜์˜€์„ ๊ฒฝ์šฐ์˜ ์„ฑ๋Šฅ์„ ๋ถ„์„ํ•œ๋‹ค. ๋‘ ๋ฒˆ์งธ๋กœ, ๋ถ„์‚ฐ ์ €์žฅ ์‹œ์Šคํ…œ์„ ์œ„ํ•œ ์ˆœํ™˜ ๋ถ€ํ˜ธ ๋ฐ ๊ธฐ์กด์˜ ๋ถ€๋ถ„ ์ ‘์† ๋ณต๊ตฌ ๋ถ€ํ˜ธ (LRC)๋ฅผ ์ด์šฉํ•œ ์ด์ง„ ํ˜น์€ ์‚ผ์ง„ ๋ถ€๋ถ„ ์ ‘์† ๋ณต๊ตฌ ๋ถ€ํ˜ธ ์„ค๊ณ„๋ฒ•์„ ์ œ์•ˆํ•˜์˜€๋‹ค. ์ดˆ๊ธฐ ์—ฐ๊ตฌ๋กœ์„œ, ์ˆœํ™˜ ๋ถ€ํ˜ธ ๋ฐ ์—ฐ์ ‘์„ ํ™œ์šฉํ•œ ์ด์ง„ ๋ฐ ์‚ผ์ง„ LRC ์„ค๊ณ„ ๊ธฐ๋ฒ•์ด ์—ฐ๊ตฌ๋˜์—ˆ๋‹ค. ์ตœ์†Œ ํ•ด๋ฐ ๊ฑฐ๋ฆฌ๊ฐ€ 4,5, ํ˜น์€ 6์ธ ์ œ์•ˆ๋œ ์ด์ง„ LRC ์ค‘ ์ผ๋ถ€๋Š” ์ƒํ•œ๊ณผ ๋น„๊ตํ•ด ๋ณด์•˜์„ ๋•Œ ์ตœ์  ์„ค๊ณ„์ž„์„ ์ฆ๋ช…ํ•˜์˜€๋‹ค. ๋˜ํ•œ, ๋น„์Šทํ•œ ๋ฐฉ๋ฒ•์„ ์ ์šฉํ•˜์—ฌ ์ข‹์€ ํŒŒ๋ผ๋ฏธํ„ฐ์˜ ์‚ผ์ง„ LRC๋ฅผ ์„ค๊ณ„ํ•  ์ˆ˜ ์žˆ์—ˆ๋‹ค. ๊ทธ ์™ธ์— ๊ธฐ์กด์˜ LRC๋ฅผ ํ™œ์šฉํ•˜์—ฌ ํฐ ํ•ด๋ฐ ๊ฑฐ๋ฆฌ์˜ ์ƒˆ๋กœ์šด LRC๋ฅผ ์„ค๊ณ„ํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์•ˆํ•˜์˜€๋‹ค. ์ œ์•ˆ๋œ LRC๋Š” ๋ถ„๋ฆฌ๋œ ๋ณต๊ตฌ ๊ตฐ ์กฐ๊ฑด์—์„œ ์ตœ์ ์ด๊ฑฐ๋‚˜ ์ตœ์ ์— ๊ฐ€๊นŒ์šด ๊ฐ’์„ ๋ณด์˜€๋‹ค. ๋งˆ์ง€๋ง‰์œผ๋กœ, GRP LDPC ๋ถ€ํ˜ธ๋Š” Nakagami-mm ๋ธ”๋ก ํŽ˜์ด๋”ฉ ๋ฐ ๋ธ”๋ก ๊ฐ„์„ญ์ด ์žˆ๋Š” ๋‘ ์ƒํƒœ์˜ ์ด์ง„ ๋Œ€์นญ ์ฑ„๋„์„ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•œ๋‹ค. ์ด๋Ÿฌํ•œ ์ฑ„๋„ ํ™˜๊ฒฝ์—์„œ GRP LDPC ๋ถ€ํ˜ธ๋Š” ํ•˜๋‚˜์˜ ๋ธ”๋ก ๊ฐ„์„ญ์ด ๋ฐœ์ƒํ–ˆ์„ ๊ฒฝ์šฐ, ์ด๋ก ์  ์„ฑ๋Šฅ์— ๊ฐ€๊นŒ์šด ์ข‹์€ ์„ฑ๋Šฅ์„ ๋ณด์—ฌ์ค€๋‹ค. ์ด๋Ÿฌํ•œ ์ด๋ก  ๊ฐ’์€ ์ฑ„๋„ ๋ฌธํ„ฑ๊ฐ’์ด๋‚˜ ์ฑ„๋„ outage ํ™•๋ฅ ์„ ํ†ตํ•ด ๊ฒ€์ฆํ•  ์ˆ˜ ์žˆ๋‹ค. ์ œ์•ˆ๋œ ์„ค๊ณ„์—์„œ๋Š”, ๋ณ€ํ˜•๋œ PEXIT ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ํ™œ์šฉํ•˜์—ฌ ๊ธฐ์ดˆ ํ–‰๋ ฌ์„ ์„ค๊ณ„ํ•œ๋‹ค. ๋˜ํ•œ AJ-PR LDPC ๋ถ€ํ˜ธ๋Š” ์ฃผํŒŒ์ˆ˜ ๋„์•ฝ ํ™˜๊ฒฝ์—์„œ ๋ฐœ์ƒํ•˜๋Š” ์ถ”์  ์žฌ๋ฐ์ด ์žˆ๋Š” ํ™˜๊ฒฝ์„ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•œ๋‹ค. ์ฑ„๋„ ํ™˜๊ฒฝ์œผ๋กœ MFSK ๋ณ€๋ณต์กฐ ๋ฐฉ์‹์˜ ๋ ˆ์ผ๋ฆฌ ๋ธ”๋ก ํŽ˜์ด๋”ฉ ๋ฐ ๋ฌด์ž‘์œ„ํ•œ ์ง€์† ์‹œ๊ฐ„์ด ์žˆ๋Š” ์žฌ๋ฐ ํ™˜๊ฒฝ์„ ๊ฐ€์ •ํ•œ๋‹ค. ์ด๋Ÿฌํ•œ ์žฌ๋ฐ ํ™˜๊ฒฝ์œผ๋กœ ์ตœ์ ํ™”ํ•˜๊ธฐ ์œ„ํ•ด, ๋ถ€๋ถ„ ๊ท ์ผ ๊ตฌ์กฐ ๋ฐ ํ•ด๋‹น๋˜๋Š” ๋ฐ€๋„ ์ง„ํ™” (density evolution, DE) ๊ธฐ๋ฒ•์ด ํ™œ์šฉ๋œ๋‹ค. ์—ฌ๋Ÿฌ ์‹œ๋ฎฌ๋ ˆ์ด์…˜ ๊ฒฐ๊ณผ๋Š” ์ถ”์  ์žฌ๋ฐ์ด ์กด์žฌํ•˜๋Š” ํ™˜๊ฒฝ์—์„œ ์ œ์•ˆ๋œ ๋ถ€ํ˜ธ๊ฐ€ 802.16e์— ์‚ฌ์šฉ๋˜์—ˆ๋˜ LDPC ๋ถ€ํ˜ธ๋ณด๋‹ค ์„ฑ๋Šฅ์ด ์šฐ์ˆ˜ํ•จ์„ ๋ณด์—ฌ์ค€๋‹ค.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 โ‰ฅ 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
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