On Decoding of the (89, 45, 17) Quadratic Residue Code

In this paper, Three decoding methods of the (89, 45, 17) binary quadratic residue (QR) code to be presented are hard, soft and linear programming decoding algorithms. Firstly, a new hybrid algebraic decoding algorithm for the (89, 45, 17) QR code is proposed. It uses the Laplace formula to obtain t...

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Published in:	IEEE transactions on communications Vol. 61; no. 3; pp. 832 - 841
Main Authors:	Wang, Lin, Li, Yong, Truong, Trieu-Kien, Lin, Tsung-Ching
Format:	Journal Article
Language:	English
Published:	New York, NY IEEE 01.03.2013 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Algebra Algorithm design and analysis Algorithms Applied sciences Approximation algorithms Berlekamp-Massey algorithm Chase algorithm Codes Coding, codes Complexity Complexity theory Decoding Exact sciences and technology Gaussian elimination Information, signal and communications theory Linear programming Maximum likelihood decoding Noise levels Performance enhancement Polynomials quadratic residue code Residues Signal and communications theory Telecommunications and information theory Performance evaluation Berlekamp-Massey algorithm Error estimation Bit error rate Error rate Berlekamp Massey algorithm Linear programming Decoding Chase algorithm Simulation Coding Gaussian elimination quadratic residue code Residue codes Algorithm complexity Error correction Soft decision
ISSN:	0090-6778, 1558-0857
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Abstract	In this paper, Three decoding methods of the (89, 45, 17) binary quadratic residue (QR) code to be presented are hard, soft and linear programming decoding algorithms. Firstly, a new hybrid algebraic decoding algorithm for the (89, 45, 17) QR code is proposed. It uses the Laplace formula to obtain the primary unknown syndromes, as done in Lin et al.'s algorithm when the number of errors v is less than or equal to 5, whereas Gaussian elimination is adopted to compute the unknown syndromes when v ≥ 6. Secondly, an appropriate modification to the algorithm developed by Chase is also given in this paper. Therefore, combining the proposed algebraic decoding algorithm with the modified Chase-II algorithm, called a new soft-decision decoding algorithm, becomes a complete soft decoding of QR codes. Thirdly, in order to further improve the error-correcting performance of the code, linear programming (LP) is utilized to decode the (89, 45, 17) QR code. Simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with Lin et al.'s hard decoding algorithm, and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate (BER) performance. Moreover, the LP-based decoding improves the error-rate performance almost without increasing the decoding complexity, when compared with the new soft-decision decoding algorithm. It provides a coding gain of 0.2 dB at BER = 2 × 10 -6 .
AbstractList	In this paper, Three decoding methods of the (89, 45, 17) binary quadratic residue (QR) code to be presented are hard, soft and linear programming decoding algorithms. Firstly, a new hybrid algebraic decoding algorithm for the (89, 45, 17) QR code is proposed. It uses the Laplace formula to obtain the primary unknown syndromes, as done in Lin et al.'s algorithm when the number of errors v is less than or equal to 5, whereas Gaussian elimination is adopted to compute the unknown syndromes when v greater than or equal to 6. Secondly, an appropriate modification to the algorithm developed by Chase is also given in this paper. Therefore, combining the proposed algebraic decoding algorithm with the modified Chase-II algorithm, called a new soft-decision decoding algorithm, becomes a complete soft decoding of QR codes. Thirdly, in order to further improve the error-correcting performance of the code, linear programming (LP) is utilized to decode the (89, 45, 17) QR code. Simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with Lin et al.'s hard decoding algorithm, and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate (BER) performance. Moreover, the LP-based decoding improves the error-rate performance almost without increasing the decoding complexity, when compared with the new soft-decision decoding algorithm. It provides a coding gain of 0.2 dB at BER = 2 x 10-6}. In this paper, Three decoding methods of the (89, 45, 17) binary quadratic residue (QR) code to be presented are hard, soft and linear programming decoding algorithms. Firstly, a new hybrid algebraic decoding algorithm for the (89, 45, 17) QR code is proposed. It uses the Laplace formula to obtain the primary unknown syndromes, as done in Lin et al.'s algorithm when the number of errors v is less than or equal to 5, whereas Gaussian elimination is adopted to compute the unknown syndromes when v ≥ 6. Secondly, an appropriate modification to the algorithm developed by Chase is also given in this paper. Therefore, combining the proposed algebraic decoding algorithm with the modified Chase-II algorithm, called a new soft-decision decoding algorithm, becomes a complete soft decoding of QR codes. Thirdly, in order to further improve the error-correcting performance of the code, linear programming (LP) is utilized to decode the (89, 45, 17) QR code. Simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with Lin et al.'s hard decoding algorithm, and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate (BER) performance. Moreover, the LP-based decoding improves the error-rate performance almost without increasing the decoding complexity, when compared with the new soft-decision decoding algorithm. It provides a coding gain of 0.2 dB at BER = 2 x 10^{-6}. In this paper, Three decoding methods of the (89, 45, 17) binary quadratic residue (QR) code to be presented are hard, soft and linear programming decoding algorithms. Firstly, a new hybrid algebraic decoding algorithm for the (89, 45, 17) QR code is proposed. It uses the Laplace formula to obtain the primary unknown syndromes, as done in Lin et al.'s algorithm when the number of errors v is less than or equal to 5, whereas Gaussian elimination is adopted to compute the unknown syndromes when v ≥ 6. Secondly, an appropriate modification to the algorithm developed by Chase is also given in this paper. Therefore, combining the proposed algebraic decoding algorithm with the modified Chase-II algorithm, called a new soft-decision decoding algorithm, becomes a complete soft decoding of QR codes. Thirdly, in order to further improve the error-correcting performance of the code, linear programming (LP) is utilized to decode the (89, 45, 17) QR code. Simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with Lin et al.'s hard decoding algorithm, and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate (BER) performance. Moreover, the LP-based decoding improves the error-rate performance almost without increasing the decoding complexity, when compared with the new soft-decision decoding algorithm. It provides a coding gain of 0.2 dB at BER = 2 × 10 -6 .
Author	Trieu-Kien Truong Lin Wang Tsung-Ching Lin Yong Li
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Keywords	Performance evaluation Berlekamp-Massey algorithm Error estimation Bit error rate Error rate Berlekamp Massey algorithm Linear programming Decoding Chase algorithm Simulation Coding Gaussian elimination quadratic residue code Residue codes Algorithm complexity Error correction Soft decision
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SubjectTerms	Algebra Algorithm design and analysis Algorithms Applied sciences Approximation algorithms Berlekamp-Massey algorithm Chase algorithm Codes Coding, codes Complexity Complexity theory Decoding Exact sciences and technology Gaussian elimination Information, signal and communications theory Linear programming Maximum likelihood decoding Noise levels Performance enhancement Polynomials quadratic residue code Residues Signal and communications theory Telecommunications and information theory
Title	On Decoding of the (89, 45, 17) Quadratic Residue Code
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