List Decoding of Generalized Reed-Solomon Codes by Using a Modified Extended Key Equation Algorithm
© Ta-Hsiang Hu and Ming-Hua Chang. 2011
Received: 27 October 2010
Accepted: 7 January 2011
Published: 16 January 2011
This work presents a modified extended key equation algorithm in list decoding of generalized Reed-Solomon (GRS) codes. A list decoding algorithm of generalized Reed-Solomon codes has two steps, interpolation and factorization. The extended key equation algorithm (EKE) is an interpolation-based approach with a lower complexity than Sudan's algorithm. To increase the decoding speed, this work proposes a modified EKE algorithm to perform codeword checking prior to such an interpolation process. Since the evaluation mapping is engaged in encoding, a codeword is not generated systematically. Thus, the transmission information is not directly obtained from a received codeword. Therefore, the proposed algorithm undertakes a matrix operation to obtain the transmission information once a received vector has been checked to be error-free. Simulation results demonstrate that the modified EKE algorithm in list decoding of a GRS code provides low complexity, particularly at high signal-to-noise ratios.
Reed-Solomon (RS) codes are currently used in a wide variety of applications, ranging from data storage systems, mobile communications, to satellite communications. The third-generation (3G) wireless standard utilizes RS codes as outer codes. For CDMA2000 high-rate broadcast packet data air interface , they are expected to be adopted as outer codes in concatenated coding schemes for future fourth-generation (4G) wireless systems.
Algorithms for hard decision decoding of RS codes are typically classified into two well-known types, namely, syndrome-based decoding and interpolation-based decoding. Well-developed algorithms in the first category include the Peterson-Gorenstein-Zierler algorithm , Berlekamp-Massey algorithm [2, 3], Euclidean algorithm [2, 3], frequency domain algorithm [2, 3], and step-by-step algorithm [4–7]. Algorithms in the second category include the Welch-Berlekamp algorithms [8, 9] and list decoding algorithms [10–12], as Koetter-Vardy algorithm  is also a list decoding algorithm but with soft decision approaching.
Sudan's algorithm  decodes GRS codes in two steps involved, namely, interpolation and factorization. An interpolation is performed on a received word , producing a nonzero bivariate polynomial with at least points , such that and . Factorization is then performed on , yielding linear factors (or called -root polynomials) . The codewords are then generated from these distinct factors via an evaluation mapping. A decoded codeword is chosen if the Hamming distance between and is or less.
Because solving these interpolation equations of Sudan's algorithm with a naïve Gaussian elimination requires the time complexity , an EKE algorithm has been presented to decrease this complexity . The EKE algorithm employs generalized Berlekamp-Massy algorithm (or the Feng-Tzeng algorithm in ) that obtains the shortest recurrence that generates a given sequence, and the time complexity of EKE to solve these interpolation equations is . represents a design parameter, typically a small constant, which is an upper bound on the size of the list of decoded codewords.
Guruswami and Sudan (GS) presented an improvement on Sudan's algorithm , by introducing a multiplicity at each interpolation point. A nonzero polynomial exists that interpolates the points , with multiplicity , and is formed by , where , and the expression of denotes the number of ways to choose from . In comparison with Sudan's work, the GS algorithm provides more linear homogeneous equations in interpolation, thus improving the decoding correction distance. Increasing improves the decoding performance but also increases the required complexity. The asymptotical decoding correction fraction is given by , and the code rate is given by . The increase in decoding capability is substantial, especially for low-rate GRS codes.
Koetter and Vardy  extended the GS algorithm by incorporating the soft information received from a channel into the interpolation process. With a complexity that is a polynomial of the code length, the Koetter-Vardy (KV) algorithm can achieve a substantial coding gain over the GS algorithm. For instance, at a frame-error-rate (FER) of , the KV algorithm can achieve a coding gain of about 1 dB over the GS algorithm, for a (255, 144) GRS code transmitted over an additive white Gaussian noise (AWGN) channel using 256-QAM modulation .
However, those approaches have a drawback, that is codeword checking is absent during decoding. In other words, regardless of whether the received sequence is correct or not, the decoding algorithm proceeds to decode it. This work overcomes this drawback by presenting a modified EKE algorithm with codeword checking. Additionally, a matrix operation is also proposed to obtain the transmission information from the received codeword. As in syndrome-based decoding, if the syndrome vector is all-zero, then the decoding process is terminated and the received sequence is output as a decoded codeword. The rest of this paper is organized as follows. Section 2 introduces the EKE algorithm. Section 3 then presents the modified EKE algorithm with the proposed codeword checking method and the matrix operation to obtain the transmission information from the received codeword. Finally, simulations and conclusions are presented in Section 4.
2. Extended Key Equation Algorithm
The EKE algorithm employs the Feng-Tzeng algorithm  to decode GRS codes. The dimensions of the -matrix in (11) are by . Since the Feng-Tzeng algorithm is run column by column in a matrix, therefore the column length dominates the decoding complexity. Reducing the column length lowers the complexity of locating the smallest set of linear dependent coefficients. The algorithm of  requires the solving of homogeneous linear equations in (11) and then finding the corresponding coefficients of in (14). Hence, the time complexity is , which is less than the time complexity of of Sudan's algorithm. Consequently, the EKE algorithm is more attractive than the algorithm of .
3. Modified Extended Key Equation Algorithm
Perform the factorization on the bivariate polynomial by employing the reconstruction algorithm  to find the -root polynomials .
4. Simulations and Conclusions
The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract no. NSC 97-2221-E-212-005.
- Agashe P, Rezaiifar R, Bender P: CDMA2000® high rate broadcast packet data air interface design. IEEE Communications Magazine 2004, 42(2):83-90.View ArticleGoogle Scholar
- Wicker SB: Error Control Systems for Digital Communication and Storage. Prentice Hall, New York, NY, USA; 1995.MATHGoogle Scholar
- Lin S, Costello DJ Jr.: Error Control Coding. 2nd edition. Prentice Hall, New York, NY, USA; 2004.Google Scholar
- Peterson WW, Weldon EJ: Error-Control Codes. 2nd edition. MIT Press, Cambridge, Mass, USA; 1972.Google Scholar
- Massy JL: Step-by-step decoding of Bose-Chauhuri-Hocquenghem codes. IEEE Transactions on Information Theory 1965, 11(4):580-585. 10.1109/TIT.1965.1053833View ArticleGoogle Scholar
- Wei SW, Wei CH: High-speed decoder of Reed-Solomon codes. IEEE Transactions on Communications 1993, 41(11):1588-1593. 10.1109/26.241736MATHView ArticleGoogle Scholar
- Chen TC, Wei CH, Wei SW: Step-by-step decoding algorithm for Reed-Solomon codes. IEE Proceedings Communications 2000, 147(1):8-12. 10.1049/ip-com:20000149View ArticleGoogle Scholar
- Welch L, Berlekamp ER: Error correction for algebraic block codes. U.S. Patent 4 633 470, 1983Google Scholar
- Fedorenko SV: A simple algorithm for decoding Reed-Solomon codes and its relation to the Welch-Berlekamp algorithm. IEEE Transactions on Information Theory 2005, 51(3):1196-1198. 10.1109/TIT.2004.842738MATHMathSciNetView ArticleGoogle Scholar
- Sudan M: Decoding of Reed Solomon codes beyond the error-correction bound. Journal of Complexity 1997, 13(1):180-193. 10.1006/jcom.1997.0439MATHMathSciNetView ArticleGoogle Scholar
- Guruswami V, Sudan M: Improved decoding of reed-solomon and algebraic-geometry codes. IEEE Transactions on Information Theory 1999, 45(6):1757-1767. 10.1109/18.782097MATHMathSciNetView ArticleGoogle Scholar
- Roth RM, Ruckenstein G: Efficient decoding of Reed-Solomon codes beyond half the minimum distance. IEEE Transactions on Information Theory 2000, 46(1):246-257. 10.1109/18.817522MATHMathSciNetView ArticleGoogle Scholar
- Koetter R, Vardy A: Algebraic soft-decision decoding of Reed-Solomon codes. IEEE Transactions on Information Theory 2003, 49(11):2809-2825. 10.1109/TIT.2003.819332MATHMathSciNetView ArticleGoogle Scholar
- Feng GL, Tzeng KK: A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes. IEEE Transactions on Information Theory 1991, 37(5):1274-1287. 10.1109/18.133246MATHMathSciNetView ArticleGoogle Scholar
- MacWilliams FJ, Sloane NJ: The Theory of Error Correcting Codes. North Holland, Amsterdam, The Netherlands; 1977.MATHGoogle Scholar
- McEliece RJ: The Theory of Information and Coding. 2nd edition. Cambridge University Press, Cambridge, UK; 2002.MATHView ArticleGoogle Scholar
- Pellikaan R, Wu XW: List decoding of q-ary Reed-Muller codes. IEEE Transactions on Information Theory 2004, 50(4):679-682. 10.1109/TIT.2004.825043MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.