A Lookup Table Decoding of systematic (47, 24, 11) quadratic residue code

A new decoding algorithm for the binary systematic (47, 24, 11) quadratic residue (QR) code, a code that allows error-correction of up to five errors, is presented in this paper. The key idea behind this decoding technique is based on the existence of a one-to-one mapping between the syndromes “S₁” and correctable error patterns. By looking up a pre-calculated table, this algorithm determines the locations of errors directly, thus requires no multiplication operations over a finite field. Moreover, the algorithm dramatically reduces the memory required by approximately 89%. A full search confirms that when five or less errors occur, this algorithm decodes these errors perfectly. Since the implementation is written in the C-language, it is readily adaptable for use in Digital Signal Processing (DSP) applications.     

On the decoding of the (24, 12, 8) Golay code

An improved syndrome shift-register decoding algorithm, called the syndrome-weight decoding algorithm, is proposed for decoding three possible errors and detecting four errors in the (24, 12, 8) Golay code. This method can also be extended to decode two other short codes, such as the (15, 5, 7) cyclic code and the (31, 16, 7) quadratic residue (QR) code. The proposed decoding algorithm makes use of the properties of cyclic codes, the weight of syndrome, and the syndrome decoder with a reduced-size lookup table (RSLT) in order to reduce the number of syndromes and their corresponding coset leaders. This approach results in a significant reduction in the memory requirement for the lookup table, thereby yielding a faster decoding algorithm. Simulation results show that the decoding speed of the proposed algorithm is approximately 3.6 times faster than that of the algebraic decoding algorithm.     

Algebraic decoding of the (41, 21, 9) Quadratic Residue code

In this paper, an algebraic decoding algorithm is proposed to correct all patterns of four or fewer errors in the binary (41, 21, 9) Quadratic Residue (QR) code. The technique needed here to decode the (41, 21, 9) QR code is different from the algorithms developed in [I.S. Reed, T.K. Truong, X. Chen, X. Yin, The algebraic decoding of the (41, 21, 9) Quadratic Residue code, IEEE Transactions on Information Theory 38 (1992 ) 974-986]. This proposed algorithm does not require to solve certain quadratic, cubic, and quartic equations and does not need to use any memory to store the five large tables of the fundamental parameters in GF(2²⁰) to decode this QR code. By the modification of the technique developed in [R. He, I.S. Reed, T.K. Truong, X. Chen, Decoding the (47, 24, 11) Quadratic Residue code, IEEE Transactions on Information Theory 47 (2001) 1181-1186], one can express the unknown syndromes as functions of the known syndromes. With the appearance of known syndromes, one can solve Newton’s identities to obtain the coefficients of the error-locator polynomials. Besides, the conditions for different number of errors of the received words will be derived. Computer simulations show that the proposed decoding algorithm requires about 22% less execution time than the syndrome decoding algorithm. Therefore, this proposed decoding scheme developed here is more efficient to implement and can shorten the decoding time.     

Fast, prime factor, discrete Fourier transform algorithms over GF(2) for 8 ? m ? 10

In this paper it is shown that Winograd’s algorithm for computing convolutions and a fast, prime factor, discrete Fourier transform (DFT) algorithm can be modified to compute Fourier-like transforms of long sequences of 2^m − 1 points over GF(2^m), for 8 ? m ? 10. These new transform techniques can be used to decode Reed-Solomon (RS) codes of block length 2^m − 1. The complexity of this new transform algorithm is reduced substantially from more conventional methods. A computer simulation verifies these new results.     

On the decoding of binary cyclic codes with the Newton identities

We revisit in this paper the concept of decoding binary cyclic codes with Gröbner bases. These ideas were first introduced by Cooper, then Chen, Reed, Helleseth and Truong, and eventually by Orsini and Sala. We discuss here another way of putting the decoding problem into equations: the Newton identities. Although these identities have been extensively used for decoding, the work was done manually, to provide formulas for the coefficients of the locator polynomial. This was achieved by Reed, Chen, Truong and others in a long series of papers, for decoding quadratic residue codes, on a case-by-case basis. It is tempting to automate these computations, using elimination theory and Gröbner bases.Thus, we study in this paper the properties of the system defined by the Newton identities, for decoding binary cyclic codes. This is done in two steps, first we prove some facts about the variety associated with this system, then we prove that the ideal itself contains relevant equations for decoding, which lead to formulas.Then we consider the so-called online Gröbner basis decoding, where the work of computing a Gröbner basis is done for each received word. It is much more efficient for practical purposes than preprocessing and substituting into the formulas. Finally, we conclude with some computational results, for codes of interesting length (about one hundred).