Asymptotic estimation of the fraction of errors correctable by q-ary LDPC codes

We consider an ensemble of random q -ary LDPC codes. As constituent codes, we use q -ary single-parity-check codes with d = 2 and Reed-Solomon codes with d = 3. We propose a hard-decision iterative decoding algorithm with the number of iterations of the order of the logarithm of the code length. We...

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Published in:	Problems of information transmission Vol. 46; no. 2; pp. 142 - 159
Main Authors:	Frolov, A. A., Zyablov, V. V.
Format:	Journal Article
Language:	English
Published:	Dordrecht SP MAIK Nauka/Interperiodica 01.06.2010
Subjects:	Coding Theory Communications Engineering Control Electrical Engineering Engineering Information Storage and Retrieval Networks Systems Theory Information Transmission LDPC Code Asymptotic Estimation Code Length Code Rate
ISSN:	0032-9460, 1608-3253
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Abstract	We consider an ensemble of random q -ary LDPC codes. As constituent codes, we use q -ary single-parity-check codes with d = 2 and Reed-Solomon codes with d = 3. We propose a hard-decision iterative decoding algorithm with the number of iterations of the order of the logarithm of the code length. We show that under this decoding algorithm there are codes in the ensemble with the number of correctable errors linearly growing with the code length. We weaken a condition on the vertex expansion of the Tanner graph corresponding to the code.
AbstractList	We consider an ensemble of random q -ary LDPC codes. As constituent codes, we use q -ary single-parity-check codes with d = 2 and Reed-Solomon codes with d = 3. We propose a hard-decision iterative decoding algorithm with the number of iterations of the order of the logarithm of the code length. We show that under this decoding algorithm there are codes in the ensemble with the number of correctable errors linearly growing with the code length. We weaken a condition on the vertex expansion of the Tanner graph corresponding to the code.
Author	Zyablov, V. V. Frolov, A. A.
Author_xml	– sequence: 1 givenname: A. A. surname: Frolov fullname: Frolov, A. A. email: alexey.frolov@iitp.ru organization: Kharkevich Institute for Information Transmission Problems, RAS – sequence: 2 givenname: V. V. surname: Zyablov fullname: Zyablov, V. V. organization: Kharkevich Institute for Information Transmission Problems, RAS
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References	GallagerR.G.Low-Density Parity-Check Codes1963CambridgeMIT PressTranslated under the title Kody s maloi plotnost’yu proverok na chetnost’, Moscow: Mir, 1966 ZyablovV.V.PinskerM.S.Estimation of the Error-Correction Complexity for Gallager Low-Density CodesProbl. Peredachi Inf.197511123360358.94017[Probl. Inf. Trans. (Engl. Transl.), 1975, vol. 11, no. 1, pp. 18–28] TannerR.M.A Recursive Approach to Low Complexity CodesIEEE Trans. Inform. Theory19812755335470474.9402910.1109/TIT.1981.1056404650686 Zyablov, V., Johannesson, R., Lončar, M., and Rybin, P., On the Erasure-Correcting Capabilities of Low-Complexity Decoded LDPC Codes with Constituent Hamming Codes, in Proc. 11th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT’2008), Pamporovo, Bulgaria, 2008, pp. 338–347. ZyablovV.V.JohannessonR.LončarM.Low-Complexity Error Correction of Hamming-Code-Based LDPC CodesProbl. Peredachi Inf.20094522540[Probl. Inf. Trans. (Engl. Transl.), 2009, vol. 45, no. 2, pp. 95–109] Zyablov, V., Potapov, V., and Groshev, F., Low-Complexity Error Correction in LDPC Codes with Constituent RS Codes, in Proc. 11th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT’2008), Pamporovo, Bulgaria, 2008, pp. 348–353. Miladinović, N. and Fossorier, M., Generalized LDPC Codes with Reed-Solomon and BCH Codes as Component Codes for Binary Channels, in Proc. IEEE Conf. on Global Telecommunications (GLOBECOM 2005), St. Louis, USA, 2005, pp. 591–596. BassalygoL.A.Formalization of the Problem of Complexity of Code SpecificationProbl. Peredachi Inf.19761241051060355.94013476157[Probl. Inf. Trans. (Engl. Transl.), 1976, vol. 12, no. 4, pp. 322–324] R.G. Gallager (4076_CR1) 1963 4076_CR8 4076_CR5 4076_CR4 R.M. Tanner (4076_CR2) 1981; 27 L.A. Bassalygo (4076_CR7) 1976; 12 V.V. Zyablov (4076_CR3) 2009; 45 V.V. Zyablov (4076_CR6) 1975; 11
References_xml	– reference: ZyablovV.V.JohannessonR.LončarM.Low-Complexity Error Correction of Hamming-Code-Based LDPC CodesProbl. Peredachi Inf.20094522540[Probl. Inf. Trans. (Engl. Transl.), 2009, vol. 45, no. 2, pp. 95–109] – reference: Zyablov, V., Johannesson, R., Lončar, M., and Rybin, P., On the Erasure-Correcting Capabilities of Low-Complexity Decoded LDPC Codes with Constituent Hamming Codes, in Proc. 11th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT’2008), Pamporovo, Bulgaria, 2008, pp. 338–347. – reference: GallagerR.G.Low-Density Parity-Check Codes1963CambridgeMIT PressTranslated under the title Kody s maloi plotnost’yu proverok na chetnost’, Moscow: Mir, 1966 – reference: Miladinović, N. and Fossorier, M., Generalized LDPC Codes with Reed-Solomon and BCH Codes as Component Codes for Binary Channels, in Proc. IEEE Conf. on Global Telecommunications (GLOBECOM 2005), St. Louis, USA, 2005, pp. 591–596. – reference: ZyablovV.V.PinskerM.S.Estimation of the Error-Correction Complexity for Gallager Low-Density CodesProbl. Peredachi Inf.197511123360358.94017[Probl. Inf. Trans. (Engl. Transl.), 1975, vol. 11, no. 1, pp. 18–28] – reference: Zyablov, V., Potapov, V., and Groshev, F., Low-Complexity Error Correction in LDPC Codes with Constituent RS Codes, in Proc. 11th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT’2008), Pamporovo, Bulgaria, 2008, pp. 348–353. – reference: BassalygoL.A.Formalization of the Problem of Complexity of Code SpecificationProbl. Peredachi Inf.19761241051060355.94013476157[Probl. Inf. Trans. (Engl. Transl.), 1976, vol. 12, no. 4, pp. 322–324] – reference: TannerR.M.A Recursive Approach to Low Complexity CodesIEEE Trans. Inform. Theory19812755335470474.9402910.1109/TIT.1981.1056404650686 – ident: 4076_CR4 doi: 10.1109/TURBOCODING.2008.4658693 – ident: 4076_CR8 – volume-title: Low-Density Parity-Check Codes year: 1963 ident: 4076_CR1 doi: 10.7551/mitpress/4347.001.0001 – ident: 4076_CR5 doi: 10.1109/GLOCOM.2005.1577850 – volume: 12 start-page: 105 issue: 4 year: 1976 ident: 4076_CR7 publication-title: Probl. Peredachi Inf. – volume: 27 start-page: 533 issue: 5 year: 1981 ident: 4076_CR2 publication-title: IEEE Trans. Inform. Theory doi: 10.1109/TIT.1981.1056404 – volume: 11 start-page: 23 issue: 1 year: 1975 ident: 4076_CR6 publication-title: Probl. Peredachi Inf. – volume: 45 start-page: 25 issue: 2 year: 2009 ident: 4076_CR3 publication-title: Probl. Peredachi Inf.
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Title	Asymptotic estimation of the fraction of errors correctable by q-ary LDPC codes
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