Efficient Software Method for Decoding of the (71, 36, 11) Quadratic Residue Code

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.

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 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.

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 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 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.

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.

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.

Energy consumption in Digital Signal Processing (DSP) application is a vital parameter of consideration. Mostly DSP applications deplete power. Battery constraint is important in many mobiles, processors, and sensor-related devices. The major critical operations in DSP architecture are multiplication and addition. Multiplication process is performed using repetitive addition. Hence, adder is the basic component used in digital signal processor. DSP architecture is proposed with the energy-efficient goal. Traditional carry select adder contains ripple carry adder block is replaced by parallel prefix adder. By replacing that block and introducing folded tree, we can reduce power used by DSP processors. This architecture can be used in all power constrained DSP applications. In the existing system, folded tree is used in architecture to reduce the number of processing elements, and they used carry look-ahead adder to perform the internal processing element operations. In the proposed system, carry look-ahead adder is replaced by LFA and PA to harness the energy depleted in an application (e.g., DSP application). Power consumption is reduced by 12–15% as compared with the existing algorithms using Cyclone III (EP3C16F484C6).

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.

10 - Hardware and software development tools

Finite Impulse Response (FIR) Filter occupied a significant position in most of Digital Signal Processing (DSP) Applications. The behaviour of FIR filters can be realized in software for DSP applications and executed on Digital Signal Processor architectures. These kinds of digital filters consider digital components, digital input and perform mathematical computations to produce digital output. In modern DSP applications demanding area efficient architecture for DSP algorithms while optimizing throughput. This can be achieved by design of hardware architectures for DSP algorithms and apply design optimization techniques to accelerate the throughput and minimize area. This article attempts to develop hardware architectures for FIR Filter and apply the design optimization techniques present in literature. In this article, High Level Synthesis concepts were adapted to develop hardware architecture for FIR filter and optimization techniques have been applied to optimize design attributes like area, throughput and power. Finally, design attributes are summarized after applying optimization techniques and published significant architectural attributes suitable for real life DSP applications.

The rapid increase in digital signal processing (DSP) applications has generated a strong demand for electrical engineers with DSP backgrounds; however, the gap between industry needs and university curricula still exists. To answer this challenge, a sequence of innovative DSP courses that emphasize hands-on experiments and practical applications were developed for continuing education in electrical and computer engineering. These courses are taught in the evening for night-school students having at least three years of work experience. These courses enable students to experiment with sophisticated DSP applications to augment the theoretical, conceptual, and analytical materials provided in traditional DSP courses. The inclusion of both software and hardware developments allows students to undertake a wide range of DSP projects for real-world applications. Assessment data concludes that the digital signal processor fundamentals course can increase learning interest and overcome the prerequisite problem of DSP laboratory experiments. This paper also briefly introduces representative examples of some challenging DSP applications.