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On Construction of Rate-Compatible Low-Density Parity-Check (RC-LDPC) Codes by Mohammadreza Yazdani and Amir H. Banihashemi Department of Systems and Computer Engineering Carleton University Ottawa, Ontario, Canada

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2 Outline Introduction and Motivation Introduction and Motivation Design Guidelines for Irregular RC-LDPC Codes Design Guidelines for Irregular RC-LDPC Codes Construction of RC-LDPC Codes Construction of RC-LDPC Codes Performance of Constructed Codes in Type-II Hybrid-ARQ Schemes Performance of Constructed Codes in Type-II Hybrid-ARQ Schemes Concluding Remarks Concluding Remarks

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3 Introduction and Motivation Rate-Compatible Codes: Rate-Compatible Codes: Different quality of service and protection requirements in packet data communications Different quality of service and protection requirements in packet data communications Adaptive coding and/or unequal error protection Adaptive coding and/or unequal error protection Type-II Hybrid-ARQ Protocols Type-II Hybrid-ARQ Protocols Single encoder/decoder pair Single encoder/decoder pair

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4 Background RC convolutional and block codes (since 1970’s) RC convolutional and block codes (since 1970’s) RC punctured turbo codes (Barbulescu and Pietrobon, 1995; Mantha and Kschischang, 1999; Rowitch and Milstein, 2000; Kim and Struber, 2000; Abou-El-Azm, El-Fishawy, Mohammed, 2000; Chundi, Yonghui and Yuezu, 2002; Babich, Montorsi and Vatta, 2002; Chan and Modestino, 2003) RC punctured turbo codes (Barbulescu and Pietrobon, 1995; Mantha and Kschischang, 1999; Rowitch and Milstein, 2000; Kim and Struber, 2000; Abou-El-Azm, El-Fishawy, Mohammed, 2000; Chundi, Yonghui and Yuezu, 2002; Babich, Montorsi and Vatta, 2002; Chan and Modestino, 2003) RC-LDPC codes (Li and Narayanan, 2002; Ha and McLaughlin, 2003) RC-LDPC codes (Li and Narayanan, 2002; Ha and McLaughlin, 2003)

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5 Irregular RC-LDPC Codes Finite block lengths Finite block lengths Puncturing and extending Puncturing and extending New structure for extensions (modular and deterministic) New structure for extensions (modular and deterministic) Puncturing starts from lower degree nodes Puncturing starts from lower degree nodes Linear-time encodable Linear-time encodable Progressive-Edge-Growth (PEG) Algorithm with optimized symbol degree distribution Progressive-Edge-Growth (PEG) Algorithm with optimized symbol degree distribution

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6 Overview of Results Type-II hybrid ARQ: Type-II hybrid ARQ: K=1024; R=8/19,8/18, …, 8/10 K=1024; R=8/19,8/18, …, 8/10 Throughput is only about 0.7 dB away from Shannon Limit Throughput is only about 0.7 dB away from Shannon Limit Outperforms similar schemes by up to 0.5 dB Outperforms similar schemes by up to 0.5 dB

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7 Design Guidelines Finding the proper rate for the mother code Finding the proper rate for the mother code Properly extending and puncturing the mother code (to preserve both performance and linear-time encodability) Properly extending and puncturing the mother code (to preserve both performance and linear-time encodability) Construct a good linear-time encodable mother code Construct a good linear-time encodable mother code

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8 Extended codes perform better than punctured codes Extended codes perform better than punctured codes At higher rates, extended codes perform poorly At higher rates, extended codes perform poorly p1p1 p2p2 Zero LDPC mother code Sparse area p3p3 p1p1 p2p2 p3p3 Zero Rate=k / (n+p 1 +p 2 +p 3 ) RC Codes Obtained by Extending

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9 An Example Characteristics of the family of RC-LDPC codes Characteristics of the family of RC-LDPC codes k=1024 k=1024 Highest rate=8/10 Highest rate=8/10 Lowest rate=8/19 Lowest rate=8/19 Rate of mother code=8/13 Rate of mother code=8/13 Puncturing and extending profile: Puncturing and extending profile: 8/10, 8/11, 8/12 8/13 8/14, 8/15, 8/16, 8/17, 8/18, 8/19 Both the mother code and the extension matrix are constructed by linear-PEG with Both the mother code and the extension matrix are constructed by linear-PEG with Puncturing is performed on lower degree nodes Puncturing is performed on lower degree nodes ExtendingPuncturing

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11 Performance of Constructed Codes in Type- II Hybrid ARQ Protocols Throughput: Throughput: N i = The total number of bits transmitted after the ith transmission N i = The total number of bits transmitted after the ith transmission p i = Probability that decoder accepts the packet after ith transmission p i = Probability that decoder accepts the packet after ith transmission F i = Frame error rate after the ith transmission. F i = Frame error rate after the ith transmission. To increase the throughput, we need to decrease the frame error rates of RC-LDPC codes. To increase the throughput, we need to decrease the frame error rates of RC-LDPC codes.

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12 Throughput Curves

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13 Concluding Remarks Guidelines for construction of irregular RC-LDPC codes with linear-time encoding were given. Guidelines for construction of irregular RC-LDPC codes with linear-time encoding were given. Using PEG construction and a modular structure for the extended codes, RC-LDPC codes with very good performance and linear-time encoding were constructed. Using PEG construction and a modular structure for the extended codes, RC-LDPC codes with very good performance and linear-time encoding were constructed. In a type-II hybrid ARQ scheme, the constructed codes achieve a throughput which is only about 0.7dB away from Shannon Limit (k=1024, R=8/19,…,8/10). In a type-II hybrid ARQ scheme, the constructed codes achieve a throughput which is only about 0.7dB away from Shannon Limit (k=1024, R=8/19,…,8/10).

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