Abstract
The concept and characterization of ε-best rational approximations (ε-BRA) are given in this paper. And, by using this concept, a decoding algorithm for some cyclic codes is presented.
The conventional algorithms (Berlekamp-Massey, Continued Fraction, Extended Euclidean, ...) allows us to correct up to e BCH≤d−1/2 errors where d is the designed minimum distance of the cyclic code. However our algorithm will be able to correct more than d−1/2 errors in case that the true distance δ be greater than d.
The Expurged Golay Code is a very good example of the algorithm presented which allows us to correct up to three errors. This code G(23,11) is 3-error correcting but, by using the conventional algorithms we can only correct up to two errors.
This work was partially supported by Spanish Grant TIC91-0472.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
E.R.Berlekamp, Algebraic Coding Theory, McGraw-Hill, 1968.
P.Bours, J.C.M.Janssen, M. Van Asperdt, H.C.A.Van Tilborg, “Algebraic. Decoding Beyond e BCH of Some Binary Cyclic Codes, when e > e BCH, IEEE Trans., IT-36, pag. 214–222, 1990.
O.Egecioglu, Ç.K.Koç, J.Rifà, “ Fast computation of continued fractions”, Computer Math. Applic. vol 21, n. 2–3, pag. 167–169, 1991.
M.Elia, “Algebraic Decoding of the (23,12,7) Golay Code”, IEEE Transactions on Information Theory, IT-33, n. 1, January 1987.
G.Feng, K.K.Tzeng, “Decoding Cyclic Codes up to Actual Minimum Distance Using Nonrecurrent Syndrome Dependence Relations”, IEEE Trans., IT-37, pag. 1716–1723, 1991.
J.H. van Lint, R.M.Wilson, “On the Minimum Distance of Cyclic Codes”, IEEE TRans. IT-32, pag. 23–41, 1986.
F.J.MacWilliams, N.J.A.Sloane. The Theory of Error-Correcting Code, North-Holland Publishing Company, 1977.
J.L.Massey, “Shift-register synthesis and BCH decoding”, IEEE Trans., IT-15, pag. 122–127, 1969.
W.W.Peterson, “Encoding and error-correction procedures for Bose-Chaudhuri codes”, IEEE Trans. Info. Theory, IT-6, pag. 459–470, 1960.
I.S.Reed, R.A.Scholtz, “The fast decoding of Reed Solomon Codes using Fermat theoretic Transforms and Continued Fractions”, IEEE Trans. Inform. Theory, vol IT.24 pag. 100–106, 1978.
C. Roos, “A New Lower Bound for the Minimum Distance of a Cyclic Code”, IEEE Trans. IT-37, pag. 330–332, 1983.
Y.Sugiyama, M.Kasahara, S.Hirasawa, T.Namekawa, “ A method for solving key equation for decoding Goppa codes”, Inform. Contr., 21, pag. 87–99, 1975.
LL.R.Welch and R.A.Scholtz, “Continued Fractions and Berlekamp's Algorithm”, IEEE Trans. Inform. Theory, vol. IT-25 No.1, pag. 19–27, 1979
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Coma, J.R. (1994). Decoding a bit more than the BCH bound. In: Cohen, G., Litsyn, S., Lobstein, A., Zémor, G. (eds) Algebraic Coding. Algebraic Coding 1993. Lecture Notes in Computer Science, vol 781. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57843-9_30
Download citation
DOI: https://doi.org/10.1007/3-540-57843-9_30
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-57843-7
Online ISBN: 978-3-540-48357-1
eBook Packages: Springer Book Archive