IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 8, AUGUST 2015 1279 Fast Decoding of the (47, 24, 11) Quadratic Residue Code Without Determining the Unknown Syndromes Pengwei Zhang, Yong Li, Member, IEEE, Hsin-Chiu Chang, Member, IEEE, Hongqing Liu, Member, IEEE, and Trieu-Kien Truong, Life Fellow, IEEE Abstract—In this paper, a hard-decision (HD) scheme is pre- sented to facilitate faster decoding of the (47, 24, 11) quadratic residue (QR) code. The new HD algorithm uses the previous scheme of decoding the (47, 24, 11) QR code up to three errors, but corrects four and five errors with new different methods. In the four-error case, the new algorithm directly determines the co- efficients of the error-locator polynomial by eliminating unknown syndromes in Newton identities and simplifies the condition that exactly indicates the occurrence of four errors. Subsequently, the reliability-based shift-search algorithm can be utilized to decode weight-5 error patterns. In other words, a five-error case can be decoded in terms of a four-error case after inverting an incorrect bit of the received word. Simulation results show that the new HD algorithm not only significantly reduces the decoding complexity in terms of CPU time but also saves a lot of memory while maintaining the same error-rate performance. Index Terms—Quadratic residue code, hard-decision decoding, unknown syndrome, threshold. I. I NTRODUCTION T HE binary quadratic residue (QR) codes introduced by Prange [1] are a nice family of cyclic BCH codes, which have code rates greater than or equal to 1/2 and generally have large minimum distances so that most of the known QR codes are the best-known codes. In the past decades, the most widely used methods for decoding binary QR codes are the Sylvester resultants [2]–[4] or Gröbner basis methods [6]. These methods can be utilized to solve the Newton identities and thus deter- mine the error-locator polynomial. However, Newton identities are nonlinear and multivariate equations with high degree, so the calculations of identities require very high complexity when the weight of the occurred error pattern becomes large. Manuscript received April 27, 2015; revised May 27, 2015; accepted May 28, 2015. Date of publication June 2, 2015; date of current version August 10, 2015. This work was supported in part by China NSF under Grant No. 61401050, Program for Chang Changjiang Scholars and Innovative Research Team in Uni- versity (IRT1299), the special fund of Chongqing Key Laboratory, the Ministry of Education Scientific Research Foundation for Returned Overseas Chinese under Grant No. F201405, Foundation and Advanced Research Projects of Chongqing Municipal Science and Technology Commission under Grants cstc2014jcyjA40027 and cstc2014jcyjA40017, Science and Technology Project of Chongqing Municipal Education Commission under Grant KJ1400425, and in part by Taiwan NSF under Grants No. NSC102-2215-M-214-003-MY2. The associate editor coordinating the review of this paper and approving it for publication was H. Saeedi. (Corresponding author: Yong Li.) P. Zhang, Y. Li, and H. Liu are with the Key Laboratory of Mobile Communication, Chongqing University of Posts and Telecommunications, Chongqing 400065, China (e-mail: zpw546@outlook.com; yongli@cqupt. edu.cn; hongqingliu@cqupt.edu.cn). H. C. Chang is with the Department of Information Engineering, I-Shou University, Kaohsiung 84001, Taiwan (e-mail: newballch@gmail.com). T. K. Truong is with the Department of Information Engineering, I-Shou Uni- versity, Kaohsiung 84001, Taiwan, and also with the Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (e-mail: truong@isu.edu.tw). Digital Object Identifier 10.1109/LCOMM.2015.2440263 In 2001, a new technique to express unknown syndromes as functions of known syndromes was proposed by He et al. [2] and the (47, 24, 11) QR code was successfully decoded up to five errors by determining the unknown syndrome S 5 . There- after, by computing the needed consecutive syndromes with the method in [2], Chang et al. [10] developed a new algebraic decoding algorithm for QR codes based on the inverse-free Berlekamp-Massey (BM) algorithm [7]–[9]. Recently, Dubney et al. [12] modified the algorithm in [2] by inverting an incorrect bit of the received word according to the bit-error probabilities given in [11] and conducting the decoding in terms of a four-error case when five errors occur thereby reducing the decoding time. More recently, Lin et al. [5] found that the correct codeword cannot be obtained directly from the error-locator polynomial given in [2] for the five-error case and re-derived the corresponding coefficients. Furthermore, Lin et al. derived the new conditions for different number of errors. However, checking the condition for the four- error case consumed a lot of time. In this paper, we develop an improved HD algorithm to decode the (47, 24, 11) QR code. It follows the previous method to correct up to three errors, but treats four and five errors with new different schemes. Firstly, this new algorithm directly obtains the error-locator polynomial by solving the Newton identities without determining the unknown syndrome S 5 in the four-error case and simplifies the condition which verifies the occurrence of four errors. Secondly, it deals with the five-error case in terms of four errors by inverting one of the received wrong bits. In particular, bit reliability [13] which is defined by the magnitude of the corresponding channel observation instead of bit-error probability in [12] is introduced to speed up the decoding in the five-error case. Simulation results show that new algorithm approximately accelerates the decoding 4.0 and 5.0 times compared with Lin et al.’s algorithm when four and five errors occur, respectively. The remainder of this article is organized as follows: The back- ground of the (47, 24, 11) QR code is reviewed in Section II. Section III proposes a new HD decoding algorithm. Simulation results are presented in Section IV. Finally, this paper concludes with a brief summary in the final section. II. TERMINOLOGY AND BACKGROUND OF QR CODES Let the length of a QR code n be a prime number of the form n = 8l ± 1, where l is a positive integer. The set Q n of quadratic residues modulo n denotes the set of nonzero squares modulo n, i.e., Q n ={i|i ≡ j 2 mod n for 1 ≤ j ≤ n − 1}. (1) Let m be the smallest positive integer such that n divides 2 m − 1. For the (47, 24, 11) QR code, m is equal to 23. Let α ∈ GF(2 23 ) be a root of the primitive polynomial x 23 + x 5 + 1. 1558-2558 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.