2,316 research outputs found

    Binary and Ternary Quasi-perfect Codes with Small Dimensions

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    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

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    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 ฮปโˆฃqsmโˆ’1\lambda|q^{sm}-1 and ฮป<(qsโˆ’1)2(qโˆ’1)2(1+ฯต)\lambda <\sqrt{\frac{(q^s-1)}{2(q-1)^2(1+\epsilon)}}, an infinite family of distance-optimal qq-ary cyclic sum-rank codes with the block length t=qsmโˆ’1ฮปt=\frac{q^{sm}-1}{\lambda}, the matrix size sร—ss \times s, the cardinality qs2tโˆ’s(2m+3)q^{s^2t-s(2m+3)} and the minimum sum-rank distance four is constructed. 2) Block length q4โˆ’1q^4-1 and the matrix size 2ร—22 \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 q4โˆ’1>>qโˆ’1q^4-1>>q-1. 3) For given positive integers mm satisfying 2โ‰คm2 \leq m, an infinite family of quasi-perfect sum-rank codes with the matrix size 2ร—m2 \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 qq-ary codes with the block lengths up to q2q^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 q2q^2 were include

    Deterministic Constructions of Binary Measurement Matrices from Finite Geometry

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    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 โ„“1\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 โ„“0\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

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    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

    ์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

    Commutative association schemes

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    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

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    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
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