4,519 research outputs found

    Polar codes for non-identically distributed channels

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    Polar codes for non-identically distributed channels

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    General Strong Polarization

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    Arikan's exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix MM, a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the {\em polarization} of an associated [0,1][0,1]-bounded martingale, namely its convergence in the limit to either 00 or 11. Arikan showed polarization of the martingale associated with the matrix G2=(1011)G_2 = \left(\begin{matrix} 1& 0 1& 1\end{matrix}\right) to get capacity achieving codes. His analysis was later extended to all matrices MM that satisfy an obvious necessary condition for polarization. While Arikan's theorem does not guarantee that the codes achieve capacity at small blocklengths, it turns out that a "strong" analysis of the polarization of the underlying martingale would lead to such constructions. Indeed for the martingale associated with G2G_2 such a strong polarization was shown in two independent works ([Guruswami and Xia, IEEE IT '15] and [Hassani et al., IEEE IT '14]), resolving a major theoretical challenge of the efficient attainment of Shannon capacity. In this work we extend the result above to cover martingales associated with all matrices that satisfy the necessary condition for (weak) polarization. In addition to being vastly more general, our proofs of strong polarization are also simpler and modular. Specifically, our result shows strong polarization over all prime fields and leads to efficient capacity-achieving codes for arbitrary symmetric memoryless channels. We show how to use our analyses to achieve exponentially small error probabilities at lengths inverse polynomial in the gap to capacity. Indeed we show that we can essentially match any error probability with lengths that are only inverse polynomial in the gap to capacity.Comment: 73 pages, 2 figures. The final version appeared in JACM. This paper combines results presented in preliminary form at STOC 2018 and RANDOM 201

    Polar Codes for Arbitrary DMCs and Arbitrary MACs

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    Polar codes are constructed for arbitrary channels by imposing an arbitrary quasigroup structure on the input alphabet. Just as with "usual" polar codes, the block error probability under successive cancellation decoding is o(2โˆ’N1/2โˆ’ฯต)o(2^{-N^{1/2-\epsilon}}), where NN is the block length. Encoding and decoding for these codes can be implemented with a complexity of O(NlogโกN)O(N\log N). It is shown that the same technique can be used to construct polar codes for arbitrary multiple access channels (MAC) by using an appropriate Abelian group structure. Although the symmetric sum capacity is achieved by this coding scheme, some points in the symmetric capacity region may not be achieved. In the case where the channel is a combination of linear channels, we provide a necessary and sufficient condition characterizing the channels whose symmetric capacity region is preserved by the polarization process. We also provide a sufficient condition for having a maximal loss in the dominant face.Comment: 32 pages, 1 figure. arXiv admin note: text overlap with arXiv:1112.177

    ๋น„๋™์ผ ๋ถ„ํฌ ๋ณ‘๋ ฌ ์ฑ„๋„์„ ์œ„ํ•œ ํด๋ผ ๋ถ€ํ˜ธ ๊ธฐ๋ฒ•๊ณผ ์ธ๋ฑ์Šค ์ฝ”๋“œ๋ฅผ ์œ„ํ•œ ์—ฐ๊ณ„ ํด๋ผ ๋ถ€ํ˜ธ ์„ค๊ณ„ ๊ธฐ๋ฒ•

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    ํ•™์œ„๋…ผ๋ฌธ (๋ฐ•์‚ฌ)-- ์„œ์šธ๋Œ€ํ•™๊ต ๋Œ€ํ•™์› : ์ „๊ธฐยท์ปดํ“จํ„ฐ๊ณตํ•™๋ถ€, 2015. 8. ์ด์ •์šฐ.๋ณธ ๋…ผ๋ฌธ์€ Part I์˜ ๋น„๋™ํ˜• ๋Œ€์นญ ์ด์ง„ ์ด์‚ฐ ๋ฌด๊ธฐ์–ต ์ฑ„๋„์—์„œ ์ฑ„๋„ ์šฉ๋Ÿ‰์„ ๋‹ฌ์„ฑ ํ•˜๋Š” ํด๋ผ ๋ถ€ํ˜ธ์˜ ์„ค๊ณ„ ๊ธฐ๋ฒ• ๋ฐ ์ฆ๋ช…๊ณผ Part II์˜ ๋น„๋™ํ˜• ์ด์ง„ ๋…๋ฆฝ ๋ฌด๊ธฐ์–ต ์ฑ„๋„์—์„œ ์ธ๋ฑ์Šค ๋ถ€ํ˜ธ์™€ ํด๋ผ ๋ถ€ํ˜ธ์˜ ์—ฐ๊ณ„๊ธฐ๋ฒ•์„ ํ†ตํ•œ ์ตœ์ ์˜ ์ „์†ก๋ฅ ์„ ๋‹ฌ์„ฑํ•˜๋Š” ์—ฐ๊ณ„ ๊ธฐ๋ฒ•์— ๋Œ€ํ•œ ์„ค๊ณ„๋กœ ๊ตฌ์„ฑ๋œ๋‹ค. Part I์—์„œ๋Š” ๋จผ์ € ๊ฐ ์ฑ„๋„์˜ ํ†ต๊ณ„์  ํŠน์„ฑ์„ ๋Œ€๋ณ€ํ•˜๋Š” ์ฑ„๋„ ํŒŒ๋ผ๋ฏธํ„ฐ๊ฐ€ ๊ฒฐ์ •์ ์ธ ํ˜•ํƒœ๋กœ ๋ถ€ํ˜ธ๊ธฐ์™€ ๋ณตํ˜ธ๊ธฐ์— ์ฃผ์–ด์ง€๋Š” ๊ฒฝ์šฐ๋Œ€ ๋Œ€ํ•ด ๋‹ค๋ฃจ๋ฉฐ, ๋‘๋ฒˆ์งธ๋กœ ์ด ํŒŒ๋ผ๋ฏธํ„ฐ๋“ค์ด ๊ฒฐ์ •์ ์ด ์•„๋‹Œ ๋žœ๋คํ•œ ๊ฐ’์œผ๋กœ์จ ์ฃผ์–ด์ง€๋Š” ๊ฒฝ์šฐ์— ๋Œ€ํ•˜์—ฌ ์ ํ•ฉํ•œ ํด๋ผ ๋ถ€ํ˜ธ ๊ธฐ๋ฒ•์— ๋Œ€ํ•ด ๊ธฐ์ˆ ํ•œ๋‹ค. ํ›„์ž๋Š” ๋‹ค์‹œ ๋‘๊ฐ€์ง€์˜ ํ•˜์œ„ ๊ฒฝ์šฐ๋กœ ๋‚˜๋‰˜๋Š”๋ฐ ํ•˜๋‚˜๋Š” ๋ชจ๋“  ํŒŒ๋ผ๋ฏธํ„ฐ๋“ค์ด ๋‹จ ํ•˜๋‚˜์˜ ํ™•๋ฅ  ๋ถ„ํฌ์— ๋Œ€ํ•œ ์‹คํ˜„๊ฐ’์ธ ๊ฒฝ์šฐ์ด๊ณ , ๋˜๋‹ค๋ฅธ ํ•œ๊ฐ€์ง€๋Š” ๊ฐ ํŒŒ๋ผ๋ฏธํ„ฐ๋“ค์ด ๊ฐ๊ฐ์˜ ์„œ๋กœ ๋‹ค๋ฅธ ํ™•๋ฅ  ๋ถ„ํฌ์˜ ์‹คํ˜„๊ฐ’์ธ ๊ฒฝ์šฐ์ด๋‹ค. ํด๋ผ ๋ถ€ํ˜ธ๋ฅผ ์ด์šฉํ•˜์—ฌ ๊ฒฐ์ •์ ์ธ ๊ฒฝ์šฐ์™€ ๋žœ๋คํ•œ ์‹คํ˜„๊ฐ’์œผ๋กœ ์ฃผ์ด์ง€๋Š” ๋ชจ๋“  ๊ฒฝ์šฐ์— ๋Œ€ํ•˜์—ฌ ํ‰๊ท  ์ฑ„๋„ ์šฉ๋Ÿ‰์„ ๋‹ฌ์„ฑ ํ• ์ˆ˜์žˆ์Œ์„ ์ฆ๋ช…ํ•œ๋‹ค. ์ด์— ๋”ํ•ด ๊ฒฐ์ •์  ์ฑ„๋„ ํŒŒ๋ผ๋ฏธํ„ฐ๊ฐ€ ๊ฐ€์ •๋œ ์‹œ์Šคํ…œ์—์„œ ์ฑ„๋„ ์ž…๋ ฅ์œผ๋กœ ์‚ฌ์šฉ๋˜๋Š” ์ •๋ณด ๋ฒกํ„ฐ์˜ ์น˜ํ™˜ ์—ฐ์‚ฐ์˜ ์ค‘์š”์„ฑ์— ๋Œ€ํ•˜์—ฌ ๋…ผํ•œ๋‹ค. ์ ์ ˆํ•œ ์น˜ํ™˜ ์—ฐ์‚ฐ์„ ์ด๋ก ์  ์ƒํ•œ๊ฐ’์ธ ์ฑ„๋„์šฉ๋Ÿ‰์— ๋Œ€ํ•œ ์ˆ˜๋ ด์†๋„๋ฅผ ํ–ฅ์ƒ ์‹œํ‚ฌ์ˆ˜ ์žˆ์Œ์„ ์˜ˆ์‹œ๋ฅผ ํ†ตํ•ด ๋ณด์ด๊ณ  ํœด๋ฆฌ์Šคํ‹ฑ ์น˜ํ™˜ ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ๊ฐœ๋ฐœํ•˜์—ฌ ๋‹ฌ์„ฑ ์ „์†ก๋ฅ  ๋˜๋Š” ์‹œ์Šคํ…œ ์‹ ๋ขฐ๋„๋ฅผ ํ–ฅ์ƒ ์‹œํ‚ฌ์ˆ˜ ์žˆ์Œ์„ ๋ณด์ธ๋‹ค. Part II์—์„œ๋Š” ํด๋ผ ๋ถ€ํ˜ธ์™€ ์ธ๋ฑ์Šค ๋ถ€ํ˜ธ๋ฅผ ์ ‘ํ•ฉ์‹œ์ผœ ์ผ์ข…์˜ ์—ฐ๊ณ„๋œ ์†Œ์Šค-์ฑ„๋„ ๋ถ€ํ˜ธ ์„ค๊ณ„ ๊ธฐ๋ฒ•์„ ๊ฐœ๋ฐœํ•˜๊ณ  ์ œ์•ˆ๋œ ๊ธฐ๋ฒ•์ด ์ตœ์ ์˜ ์ „์†ก๋ฅ ์„ ๋‹ฌ์„ฑํ•จ์„ ๋ณด์ธ๋‹ค. ๋จผ์ € ์ธ๋ฑ์Šค ๋ถ€ํ˜ธ์—์„œ ์ˆ˜์‹ ๋…ธ๋“œ์—์„œ ์†ก์‹ ๋…ธ๋“œ๋กœ ์ „๋‹ฌ๋˜๋Š” ๋ถ€๊ฐ€์ •๋ณด๋ฅผ ํ†ตํ•ด ๊ทธ๋ ค์ง€๋Š” ๊ทธ๋ž˜ํ”„๊ฐ€ ์™„์ „๊ทธ๋ž˜ํ”„์ผ๋•Œ ํ•ญ์ƒ ์ตœ์ ์˜ ๋‹ฌ์„ฑ ๊ธฐ๋ฒ•์ด ์กด์žฌํ•จ์„ ๋ณด์ด๊ณ , ์ด๋ฅผ ์ž„์˜์˜ ๋ถ€๊ฐ€์ •๋ณด ํŒจํ„ด์ด ์ฃผ์–ด์ง€๋Š” ๊ฒฝ์šฐ๋กœ ํ™•์žฅํ•œ๋‹ค. ์™„์ „ ๊ทธ๋ž˜ํ”„๊ฐ€ ๊ทธ๋ ค์ง€๋Š” ๊ฒฝ์šฐ์™€ ๋‹ฌ๋ฆฌ ์ž„์˜์˜ ํŒจํ„ด์œผ๋กœ ์ฃผ์–ด์ง€๋Š” ๊ฒฝ์šฐ๋Š” ๋ถ€๊ฐ€์ •๋ณด๋“ค์ด ํŠน์ • ์กฐ๊ฑด์„ ๋งŒ์กฑํ•˜๋Š” ๊ฒฝ์šฐ์— ํ•œํ•˜์—ฌ ์ตœ์  ์ „์†ก๋ฅ ์„ ๋‹ฌ์„ฑํ•˜๊ฒŒ๋จ์„ ๋ณด์ด๊ณ  ์ด๋ฅผ ๋งŒ์กฑํ•˜๋Š” ์ธ๋ฑ์Šค-ํด๋ผ ๋ถ€ํ˜ธ ์„ค๊ณ„ ๊ธฐ๋ฒ•์„ ์ œ์•ˆํ•œ๋‹ค. ๋งˆ์ง€๋ง‰์œผ๋กœ ๋ถ€๊ฐ€์ •๋ณด๊ฐ€ ๊ฒฐ์ •์ ์œผ๋กœ ์ฃผ์–ด์ง€์ง€ ์•Š๊ณ  ์กด์žฌ์„ฑ์„ ํ‘œํ˜„ํ•˜๋Š” ํ™•๋ฅ ๋กœ์จ ์ฃผ์–ด์ง€๋Š” ๊ฒฝ์šฐ ์ œ์•ˆ๋œ ์—ฐ๊ณ„๊ธฐ๋ฒ•์„ ์ด์šฉํ•œ ํ‰๊ท  ์ „์†ก๋ฅ ์— ๋Œ€ํ•˜์—ฌ ๋…ผํ•œ๋‹ค.Abstract i Contents iv List of Figures viii List of Tables xii I Polar codes for Non i.i.d. Parallel channels 1 Chapter 1 Introduction 1.1 Backgrounds 1.2 Scope and Organization Chapter 2 Polar codes with deterministic non-identically distributed channels 2.1 Non-identical channels with deterministic CP 2.1.1 The evolution of Symmetric Capacities 2.1.2 Achievable Scheme based on the symmetric capacity 2.1.3 The evolution of Bhattacharayya Parameters 2.1.4 Supermartingale Zn 2.1.5 Convergence of Zn 2.2 Channel mapping via the Interleaver Q 2.2.1 Exhaustive Search Method with Grouping 2.2.2 Heuristic method 2.3 Link failures: Puncturing operation 2.4 Polarizations on non-independent channels 2.5 Summary Chapter 3 Non-identical Binary Erasure Channels with random Erasure probabilities with Single distribution 3.1 Non-identical Binary Erasure Channels with random Erasure probabilities with Single distribution 3.1.1 Proof of Theorem 2 3.1.2 The Achievable Polar coding scheme 3.2 Random Erasure probabilities with non-identical distributions 3.2.1 Case1: Variable coding structure 3.2.2 Case2: Fixed coding structure 3.3 Summary II Polar codes schemes for Index Coded Systems Chapter 4 Nested Polar codes structures for Index codes 4.1 Introduction to Index codes 4.2 Nested structures for NC and Polar codes 4.3 ICPC for fully connected SI 4.3.1 General channel setting 4.3.2 Degraded channel setting 4.3.3 IC gain analysis 4.4 ICPC for Arbitrary SI 4.4.1 Proof of the Lemma 6 4.4.2 Proof of the Theorem 5 4.4.3 Achievable ICPC scheme for degraded structures 4.4.4 Proof of the Corollary 2 4.4.5 The ICPC scheme 4.4.6 Example: Partially Perfect Graph 4.5 ICPC for Probabilistic Side Information 4.5.1 Random ICPC for non-identical B-DMCs 4.5.2 Expected rate maximization 4.5.3 Expected achievable rate via Random graph 4.6 Summary Chapter 5 Conclusions 121 Appendix A A.1 Proof of (2.25) A.2 Proof of (2.36) A.3 Proof of (2.37) A.4 Proof of the number of equivalent channel combinations Bibliography Abstract in Korean 138Docto

    Channel Polarization on q-ary Discrete Memoryless Channels by Arbitrary Kernels

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    A method of channel polarization, proposed by Arikan, allows us to construct efficient capacity-achieving channel codes. In the original work, binary input discrete memoryless channels are considered. A special case of qq-ary channel polarization is considered by Sasoglu, Telatar, and Arikan. In this paper, we consider more general channel polarization on qq-ary channels. We further show explicit constructions using Reed-Solomon codes, on which asymptotically fast channel polarization is induced.Comment: 5 pages, a final version of a manuscript for ISIT201
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