Fast decoding of the (47, 24, 11) Quadratic ResidueCode without determining the unknown syndromes (original) (raw)

Decoding of the Triple-Error-Correcting Binary Quadratic Residue Codes

In this paper, a more efficient syndrome-weight decoding algorithm (SWDA), called the enhanced syndrome-weight decoding algorithm (ESWDA), is presented to decode up to three possible errors for the binary systematic (23, 12, 7) and (31, 16, 7) quadratic residue (QR) codes. In decoding of the QR codes, the evaluation of the error-locator polynomial in the finite field is complicated and time-consuming. To solve such a problem, the proposed ESWDA avoids evaluating the complicated error-locator polynomial, and has no need of a look-up table to store the syndromes and their corresponding error patterns in the memory. In comparison with the SWDA developed by Lin-Chang-Lee-Truong (2010), the simulation results show that the ESWDA can serve as an efficient and high-speed decoder.

Decoding the (47,24,11) quadratic residue code using bit-error probability estimates

IEEE Transactions on Communications, 2009

A new algorithm is developed to facilitate faster decoding of the (47,24,11) Quadratic Residue (QR) code. This decoder, based on the idea first developed by Reed in a 1959 MIT Lincoln Laboratory Report, uses real channel data to estimate the individual bit-error probabilities in a received word. The algorithm then sequentially inverts the bits with the highest probability of error until one of the errors is canceled. The remaining errors are then corrected by the use of algebraic decoding techniques. This new algorithm, called the reliabilitysearch algorithm, is a complete decoder that significantly reduces the decoding complexity in terms of CPU time while maintaining the same bit-error rate (BER) performance. In fact, this algorithm is an appropriate modification to the algorithm developed by Chase.

On the Decoding of [47, 24, 11] and [48, 24, 12] Quadratic Residue Codes by Some New Fast Algorithms

2021

In this paper, a new decoding algorithm, called modified reduced lookup table decoding (MRLTD) algorithm, is proposed for decoding [47, 24, 11] quadratic residue (QR) code which needs lower memory requirement compared with the algorithm based on the full lookup table and the cyclic weight decoding algorithm (CWDA). Although the memory requirements (space complexity) of the proposed algorithm are the same with a newly optimized cyclic weight (OCW) decoding algorithm, the proposed MRLTD algorithm is faster, i.e. the running-time complexity of the proposed algorithm is lower than OCW algorithm. The idea behind this decoding technique is based on the existence of a one-to-one relation between the syndromes and correctable error patterns, reported in a lookup table containing 300 syndromes which are searched by a binary search algorithm. Moreover, by a bit modification on MRLTD algorithm, it can be applied to decode effectively the extended [47, 24, 11] QR code, i.e. [48, 24, 12] QR code, applicable for correcting five errors and detecting six errors.

Decoding of the Seven-Error-Correcting Binary Quadratic Residue Codes

2017

In this paper, fast syndrome-weight decoding algorithm (FSWDA) is proposed to decode up to seven possible errors in a binary systematic quadratic residue (QR) codes (79, 40, 15) and (97, 49, 15). The main conception of FSWDA is predicated on the property of cyclic codes together with the weight of syndrome difference. In decoding of the QR codes, the evaluation of the error-locator polynomial in the finite field is complicated and time-consuming. To deal with such an issue, our scheme FSWDA keeps away from evaluating the error locator polynomial and has no need to generate the table which store the syndromes and their corresponding patterns of error in the memory. Also, our scheme serve as an efficient and high speed decoder. AMS subject classification: 94A60.

A new fast algorithm for decoding the [ 47 , 24 , 11 ] quadratic residue code

2018

In this paper, we present a new method for decoding [47,24,11] quadratic residue (QR) code which is the reduction of the required memory compared to the full lookup table or other known algebraic decoding methods. The idea behind this decoding technique is based on the existence of a one-to-one relation between the syndrome and correctable error patterns. In this approach, errors are directly found from the table and no multiplication operation over a finite field is required. Because of using the cyclic structure of codes, weight of syndrome and a reduced lookup table, the proposed algorithm can be applied on many other cyclic codes.

High speed decoding of the binary (47,24,11) quadratic residue code

Information Sciences, 2010

In this dissertation, an efficient table lookup decoding algorithm (TLDA) is presented to correct up to five possible errors in a binary systematic (47, 24, 11) quadratic residue (QR) code. The main idea of the TLDA is based on the weight of syndrome, the syndrome decoder together with a reduced-size lookup table (RSLT), and the shift-search method given by Reed et al. Thus, the size of the lookup table and computational complexity in a finite field can be significantly reduced. The memory size of the proposed condensed lookup table (CLT) consists of only 36.6 Kbytes and is only about 0.24% of the full lookup table (FLT) and about 3.4% of the lookup table given by Chen et al., respectively. These facts lead to significant reduction of computational time and the decoding complexity. A simulation result shows that the decoding speed of the proposed TLDA is much faster than all existing decoding algorithms. Moreover, it can be extended to decode all QR codes, including the class of the cyclic codes when the code length is moderate. The CLT makes this new decoding algorithm suitable for hardware or firmware implementations.

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

IEEE Communications Letters, 2011

In this letter, the algebraic decoding algorithm of the (89, 45, 17) binary quadratic residue (QR) code proposed by Truong et al. is modified by using the efficient determination algorithm of the primary unknown syndromes. The correctness of the proposed decoding algorithm is verified by computer simulations and the use of two corollaries. Also, simulation results show that the CPU time of this algorithm is approximately 4 times faster than that of the previously mentioned decoding algorithm at least. Therefore, such a fast decoding algorithm can now be applied to achieve efficiently the reliability-based decoding for the (89, 45, 17) QR code. Finally, the performance of its algebraic soft-decision decoder expressed in terms of the bit-error probability versus / 0 is given but not available in the literature.

On Decoding of the (73, 37, 13) Quadratic Residue Code

IEEE Transactions on Communications, 2014

In this paper, a method to search the set of syndromes' indices needed in computing the unknown syndromes for the (73, 37, 13) quadratic residue (QR) code is proposed. According to the resulting index sets, one computes the unknown syndromes and thus finds the corresponding error-locator polynomial by using an inverse-free Berlekamp-Massey (BM) algorithm. Based on the modified Chase-II algorithm, the performance of soft-decision decoding for the (73, 37, 13) QR code is given. This result is new. Moreover, the error-rate performance of linear programming (LP) decoding for the (73, 37, 13) QR code is also investigated, and LP-based decoding is shown to be significantly superior in performance to the algebraic soft-decision decoding while requiring almost the same computational complexity. In fact, the algebraic hard-decision and soft-decision decoding of the (89, 45, 17) QR code outperforms that of the (73, 37, 13) QR code because the former has a larger minimal distance. However, experimental results indicate that the (73, 37, 13) QR code outperforms the (89, 45, 17) QR code with much fewer arithmetic operations when using the LP-based decoding algorithms. The pseudocodewords analysis partially explains this seemingly strange phenomenon.

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

IEEE Transactions on Communications, 2000

In this paper, a modification decoding algorithm for the (89, 45, 17) QR code is proposed to decode all error patterns with weight less than six and high percentage weight-6, weight-7, and weight-8 error patterns.