Abstract
It is shown that the generalized Berlekamp-Massey algorithm (GBMA, in short) for solving the linear synthesis problem of a multi-sequence r over F 2 can be obtained naturally from a special form of the multi-continued fraction algorithm, called the multi-strict continued fraction algorithm (m-SCFA, in short). Moreover, the discrepancy sequence in acting GBMA on r is expressed explicitly by the data associated to the multi-strict continued fraction expansion C( r ) which is obtained by applying m-SCFA on r. As a consequence, a 1-1 correspondence between multi-sequences of any given length and certain multi-strict continued fractions is established.
This work is partly supported by NSFC (Grant No. 60173016), and the National 973 Project (Grant No. 1999035804).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Mills, W.H.: Continued fractions and linear recurrences. Math. Comp. 29, 173–180 (1975)
Welch, L.R., Scholtz, R.A.: Continued fractions and Berlekamp’s algorithm. IEEE Transactions on Information Theory IT-25(1), 19–27 (1979)
Schmidt, W.M.: On continued fraction and diophantine approximation in power series fields. Acta Arith. 95, 139–166 (2000)
Berlekamp, E.R.: Algebraic coding theory. McGraw Hill, New York (1969)
Massey, J.L.: Shift register synthesis and BCH decoding. IEEE Transactions on Information Theory 15(1), 122–127 (1969)
Dai, Z., Zeng, K.: Continued fractions and Berlekamp-Massey algorithm. In: Seberry, J., Pieprzyk, J.P. (eds.) AUSCRYPT 1990. LNCS, vol. 453, pp. 24–31. Springer, Heidelberg (1990)
Sakata, S.: Extension of Berlekamp-Massey algorithm to N dimension. Inform. And Comput. 84, 207–239 (1990)
Feng, G.L., Tzeng, K.K.: A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes. IEEE Transactions on Information Theory IT-37, 1274–1287 (1991)
Dai, Z., Wang, K., Ye, D.: m-Continued Fraction Expansions of Multi-Laurent Series. Advances In Mathematics(China) 33(2), 246–248 (2004)
Dai, Z., Wang, K., Ye, D.: Multidimensional Continued Fraction and Rational Approximation (2004), http://arxiv.org/abs/math.NT/0401141
Dai, Z., Wang, K., Ye, D.: Multi-Continued Fraction Algorithm on Multi-Formal Laurent Series, pp.1–21 (preprint)
Niederreiter, H.: Some computable complexity measures for binary sequences. In: Ding, C., Helleseth, T., Niederreiter, H. (eds.) Sequences and their applications, pp. 67–78. Springer, London (1999)
Wang, L.: Lattice bases reduction algorithm in function fields and multisequence linear feedback shift-register synthesis, PhD thesis, Graduate school of Chinese Academy of Sciences, Beijing (2003)
Vielhaber, M.: A unified view on sequence complexity measures as isometries. In: Proceedings (Extended Abstracts) 2004 International Conference on Sequenes and Their Applications, pp. 34–38
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dai, Z., Feng, X., Yang, J. (2005). Multi-continued Fraction Algorithm and Generalized B-M Algorithm over F 2 . In: Helleseth, T., Sarwate, D., Song, HY., Yang, K. (eds) Sequences and Their Applications - SETA 2004. SETA 2004. Lecture Notes in Computer Science, vol 3486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11423461_25
Download citation
DOI: https://doi.org/10.1007/11423461_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26084-4
Online ISBN: 978-3-540-32048-7
eBook Packages: Computer ScienceComputer Science (R0)