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(2^2^0) 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.