Algebraic decoding of the (73, 37, 13) quadratic residue code

In this study, an efficient and fast algebraic decoding algorithm (ADA) for the binary systematic quadratic residue (QR) code of length 73 with the reducible generator polynomial to correct up to six errors is proposed. The S(I, J) matrix method given by He et al. (2001) is utilised to compute the unknown syndromes S5. A technique called swap base is proposed to correct the weight-4 error patterns. To correct the weight-5 error patterns, the new error-locator polynomials for decoding the five errors are derived. Finally, the modified shift-search algorithm (SSA) developed by Lin et al. (2010) is applied to correct the weight-6 error patterns. Moreover, the computations of all syndromes are achieved in a small finite field. Simulation results show that the proposed ADA is practical.

[1]  Trieu-Kien Truong,et al.  Decoding the (73, 37, 13) quadratic residue code , 1994 .

[2]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[3]  S. Wicker.  Error Control Systems for Digital Communication and Storage , 1994 .

[4]  Tsung-Ching Lin,et al.  High speed decoding of the binary (47, 24, 11) quadratic residue code , 2010, Inf. Sci..

[5]  Robert T. Chien,et al.  Cyclic decoding procedures for Bose- Chaudhuri-Hocquenghem codes , 1964, IEEE Trans. Inf. Theory.

[6]  Trieu-Kien Truong,et al.  Algebraic decoding of the (32, 16, 8) quadratic residue code , 1990, IEEE Trans. Inf. Theory.

[7]  Trieu-Kien Truong,et al.  VLSI Architectures for Computing Multiplications and Inverses in GF(2m) , 1983, IEEE Transactions on Computers.

[8]  Hsin-Chiu Chang,et al.  An Efficient Decoding Algorithm for the (73, 37, 13) Quadratic Residue Code , 2011 .

[9]  Xuemin Chen,et al.  The algebraic decoding of the (41, 21, 9) quadratic residue code , 1992, IEEE Trans. Inf. Theory.

[10]  Chong-Dao Lee,et al.  Algebraic decoding of (71, 36, 11), (79, 40, 15), and (97, 49, 15) quadratic residue codes , 2003, IEEE Trans. Commun..

[11]  Trieu-Kien Truong,et al.  Decoding the (24,12,8) Golay code , 1990 .

[12]  Xuemin Chen,et al.  Decoding the (47, 24, 11) quadratic residue code , 2001, IEEE Trans. Inf. Theory.

[13]  Tsung-Ching Lin,et al.  On the decoding of the (24, 12, 8) Golay code , 2010, Inf. Sci..

[14]  Tsung-Ching Lin,et al.  Decoding of the (31, 16, 7) quadratic residue code , 2010 .

[15]  Tor Helleseth,et al.  Use of Grobner bases to decode binary cyclic codes up to the true minimum distance , 1994, IEEE Trans. Inf. Theory.

[16]  Michele Elia,et al.  Algebraic decoding of the (23, 12, 7) Golay code , 1987, IEEE Trans. Inf. Theory.

[17]  Chien-Hsiang Huang,et al.  A Lookup Table Decoding of systematic (47, 24, 11) quadratic residue code , 2009, Inf. Sci..

[18]  Yan-Haw Chen,et al.  Fast algorithm for decoding of systematic quadratic residue codes , 2011, IET Commun..

[19]  Chong-Dao Lee,et al.  Algebraic decoding of (103, 52, 19) and (113, 57, 15) quadratic residue codes , 2005, IEEE Transactions on Communications.