Some ideas about fault-tolerant Chinese Remaindering

The Quadratic Residue (QR) codes have a rich mathematic structure. Unfortunately, their Algebraic Decoding (AD) is not generalizable for all QR codes. In this study, an efficient hard decoding algorithm is proposed to generalize the decoding of the binary systematic Quadratic Residue (QR) codes. The proposed decoder corrects t erroneous bits or less, in the received word, based on a reduced set of permutations derived from the large automorphism group of QR codes. This set of permutations is applied to the received word to move the error positions and trap all of them in redundancy. Then, to evaluate the proposed method, we applied it to many binary QR codes of moderate code length starting with 17 until 113 with reducible and irreducible generator polynomials. The proposed decoder was validated by inserting all possible error patterns, that have t or less erroneous positions, as input of the proposed decoder and the output is always a correct codeword. The complexity study, in terms of the number of operations used, reveals that the light permutation decoding LPD algorithm significantly decreases decoding complexity without performance loss. So, it is qualified to be a good competitor to decode QR codes with lower lengths but is the best for QR codes with higher lengths.

A method for decoding of the quadratic residue (QR) code with hash tables is presented. The method can be applied in decoding the (31, 16, 7) QR code that the generator polynomial can be factored. In other words, the mapping between elements of syndrome S1 and all correctable error patterns is not one-to-one. Therefore, the decoding of the (31, 16, 7) QR code needs to stuck S1,S5, S7 known syndrome mapping an error pattern that has one-to-one nature, where a subscript 1, 5, 7 are cyclotomic cosets. Furthermore, the algorithm determines the error locations by hash tables without operating the additions and multiplications over finite field. To decode QR code result, the hash table method for the (31, 16, 7) QR code is dramatically reduced the memory size above 97%. It is very suitable for high speed in modern communication system.

A new algebraic approach for decoding the quadratic residue (QR) codes, in particular the (41, 21, 9) QR code, is presented. The key ideas behind this decoding technique are a systematic application of the Sylvester resultant method to the Newton identities associated with the syndromes to find the error-locator polynomial, and next a method for determining error locations by solving certain quadratic, cubic, and quartic equations over GF(2/sup m/) in a new way which uses Zech's logarithms for the arithmetic. The logarithms developed for Zech's logarithms save a substantial amount of computer memory by storing only a table of Zech's logarithms. These algorithms are suitable for implementation in a programmable microprocessor or special-purpose VLSI chip. It is expected that the algebraic methods developed can apply generally to other codes such as the BCH and Reed-Solomon codes. >

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 E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is given but not available in the literature.

In the context of informatization and big data, the demands for the reliability and real-time of information transmission in wireless communication channels are increasing. Quadratic residue (QR) codes have a high prospect on error correction for reliable data conveyance over channel with noise. This paper proposes an optimized algebraic scheme for decoding (71, 36, 11) QR code for correcting up to 5 errors on one-case-by-one-case basis. This scheme is mainly to improve its algebraic decoding algorithm based on the Lin’s algorithm, only in this paper, the Lin’s algorithm is called LTC71 algorithm. The key technology is to use the algebraic structure of QR code and give a new discriminant condition by mathematical derivation, which can quickly detect whether there are four errors in the code. Simulation results show that the proposed decoding scheme achieves the same bit error-rate performance as LTC71 algorithm, and the decoding time is reduced by about 28.72% owing to the reduced complexity of the decoder when there are four errors in the code.

In this paper, an effective table lookup decoding algorithm (TLDA), called the syndrome and syndrome difference decoding algorithm (SSDDA), is developed to decode the binary systematic (41, 21, 9) quadratic residue (QR) code. It is based on the property of the weight of syndrome and the weight of syndrome difference to reduce the memory size of the lookup table. The proposed algorithm requires a lookup table, called the compact lookup table (CLT), which only consists of 231 syndromes and their corresponding error patterns. The advantage of the SSDDA over the previous TLDAs is that the memory size of the proposed CLT is only about 82.2% and 2% of the lookup table needed in the decoding algorithms of Lin et al. (2010) and Chen et al. (2010), respectively.

In this paper presents an efficient table lookup algorithm for high-throughput decoding of the (71, 36, 11) Quadratic Residue (QR) code. The main ideas behind this decoding technique are based on one-to-one mapping between the syndromes S1 and correctable error patterns. As compared with the binary lookup table method, the presented technique is faster than binary searching method for finding error pattern. In addition, it has the advantage that only the operations of addition over a Finite Field are used for decoding QR codes. Moreover, the presented high speed of the decoding procedure has potential applications in modern communication systems and digital signal processing (DSP) systems.

The techniques needed to decode the (47,24,11) quadratic residue (QR) code differ from the schemes developed for cyclic codes. By finding certain nonlinear relations between the known and unknown syndromes for this special code, two methods are developed to decode up to the true minimum distance of the (47,24,11) QR code. These algorithms can be utilized to decode effectively the 1/2 -rate (48,24,12) QR code for correcting five errors and detecting six errors.

A general algorithm for decoding the binary systematic quadratic residue (QR) codes with lookup tables is presented in this study. The algorithm can be applied in decoding the QR codes with either reducible or irreducible generator polynomials. If the generator polynomial of the QR codes is reducible, the number of elements in the Galois field is less than the sum of all correctable error patterns. In other words, the mapping between elements of syndrome set and all correctable error patterns is not one to one. The key idea of decoding based on the mapping between the ordered q-tuples of the primary known syndrome and error patterns is one to one. In addition, the algorithm directly determines the error locations by lookup tables without the operations of multiplication over a finite field. According to the simulation result, the new lookup table decoding algorithm for the (31, 16, 7) QR code and the (73, 37, 13) QR code dramatically reduces the memory required by approximately 90 and 92%, respectively. Moreover, the high speed of decoding procedure could be utilised in modern communication system.

In this paper, an algebraic decoding method is proposed for the quadratic residue codes that utilize the Berlekamp-Massey algorithm. By a modification of the technique developed by He et al., one can express the unknown syndromes as functions of the known syndromes. The unknown syndromes are determined by an efficient algorithm also developed in this paper. With the appearance of unknown syndromes, one obtains the consecutive syndromes that are needed for the application of the Berlekamp-Massey algorithm. The decoding scheme, developed here, is easier to implement than the previous decoding algorithm developed for the Golay code and the (47, 24, 11) QR code. Moreover, it can be extended to decode all codes of the family of binary quadratic residue codes with irreducible generating polynomials.

The class of binary quadratic residue (QR) codes of length n=8m-1 contains two perfect codes. These are the (7,4,3) Hamming code and the (23,12,7) Golay code. However, it is proved in the present paper that there are no quasi-perfect QR codes of length 8m-1. Finally, this result is generalized to all binary self-dual codes of length N>72. >

Binary quadratic residue (QR) codes, which have code rates greater than or equal to 1/2 and generally have large minimum distances, are among the best known codes. This paper considers a modified algebraic decoding algorithm for the (89,45,17) binary QR code that utilizes the Berlekamp-Massey algorithm. It identifies the primary unknown syndromes and provides methods to determine these on a case-by-case basis for any number of correctable errors. Numerical evaluation shows that the proposed algorithm significantly reduces at least 52% of decoding time for two or more errors.

The algebraic decoding of binary quadratic residue codes can be performed using the Peterson or the Berlekamp–Massey algorithm once certain unknown syndromes are determined. In this work, the technique of determining unknown syndromes is extended to the ternary case to decode the expurgated ternary quadratic residue code of length 37.

An algebraic decoding of the (89, 45, 17) quadratic residue code suggested by Truong et al. (2008) has been designed that uses the inverse-free Berlekamp-Massey (BM) algorithm to determine the error-locator polynomial and applies a verification method to check whether the error pattern obtained by decoding algorithm is correct or not. In this paper, based on the ideas of the algorithm mentioned above, two decoding methods of the (31, 16, 7) binary quadratic residue code are proposed. Also, the comparison of the decoding complexity in terms of CPU time of these two methods and the conventional algebraic decoding method proposed by Reed et al. (1990) are given.

An algebraic decoding algorithm for the ternary (13,7,5) quadratic residue code is presented. This seems to be the first attempt to provide an algebraic decoding algorithm for a quadratic residue code over a nonbinary field.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>