Abstract
The Berlekamp–Massey and Berlekamp–Massey–Sakata algorithms compute a minimal polynomial or polynomial set of a linearly recurring sequence or multi-dimensional array. In this paper some underlying properties of and connections between these two algorithms are clarified theoretically: a unified flow chart for both algorithms is proposed to reveal their connections; the polynomials these two algorithms maintain at each iteration are proved to be reciprocal when both algorithms are applied to the same sequence; and the uniqueness of the choices of polynomials from two critical polynomial sets in the Berlekamp–Massey–Sakata algorithm is investigated.
This work was partially supported by the National Natural Science Foundation of China (NSFC 11771034) and the Fundamental Research Funds for the Central Universities in China (YWF-19-BJ-J-324).
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Berlekamp, E.: Nonbinary BCH decoding. IEEE Trans. Inf. Theory 14(2), 242–242 (1968)
Berthomieu, J., Boyer, B., Faugère, J.C.: Linear algebra for computing Gröbner bases of linear recursive multidimensional sequences. J. Symb. Comput. 83, 36–67 (2017)
Berthomieu, J., Faugère, J.C.: In-depth comparison of the Berlekamp-Massey-Sakata and the scalar-FGLM algorithms: The adaptive variants. arXiv:1806.00978 (2018, preprint)
Berthomieu, J., Faugère, J.C.: A polynomial-division-based algorithm for computing linear recurrence relations. In: Proceedings of ISSAC 2018, pp. 79–86. ACM Press (2018)
Bras-Amorós, M., O’Sullivan, M.: The correction capability of the Berlekamp-Massey-Sakata algorithm with majority voting. Appl. Algebra Eng. Commun. Comput. 17(5), 315–335 (2006)
Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry, 2nd edn. Springer Verlag, New York (2005). https://doi.org/10.1007/b138611
Faugère, J.C., Mou, C.: Fast algorithm for change of ordering of zero-dimensional Gröbner bases with sparse multiplication matrices. In: Proceedings of ISSAC 2011, pp. 115–122. ACM, ACM Press (2011)
Faugère, J.C., Mou, C.: Sparse FGLM algorithms. J. Symb. Comput. 80(3), 538–569 (2017)
Kaltofen, E., Yuhasz, G.: On the matrix Berlekamp-Massey algorithm. ACM Trans. Algorithms 9(4), 33 (2013)
Massey, J.: Shift-register synthesis and BCH decoding. IEEE Trans. Inf. Theory 15(1), 122–127 (1969)
Matsui, H., Sakata, S., Kurihara, M., Mita, S.: Systolic array architecture implementing Berlekamp-Massey-Sakata algorithm for decoding codes on a class of algebraic curves. IEEE Trans. Inf. Theory 51(11), 3856–3871 (2005)
O’Sullivan, M.: New codes for the Berlekamp-Massey-Sakata algorithm. Finite Fields Appl. 7(2), 293–317 (2001)
Saints, K., Heegard, C.: Algebraic-geometric codes and multidimensional cyclic codes: a unified theory and algorithms for decoding using Gröbner bases. IEEE Trans. Inf. Theory 41(6), 1733–1751 (2002)
Sakata, S.: Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array. J. Symb. Comput. 5(3), 321–337 (1988)
Sakata, S.: Extension of the Berlekamp-Massey algorithm to \(N\) dimensions. Inf. Comput. 84(2), 207–239 (1990)
Sakata, S.: The BM algorithm and the BMS algorithm. In: Vardy, A. (ed.) Codes, Curves, and Signals: Common Threads in Communications. SECS, vol. 485, pp. 39–52. Springer, Boston (1998). https://doi.org/10.1007/978-1-4615-5121-8_4
Wiedemann, D.: Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theory 32(1), 54–62 (1986)
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The authors would like to thank the anonymous reviewers for their helpful suggestion which contribute to considerable improvement of this paper.
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Mou, C., Fan, X. (2019). On Berlekamp–Massey and Berlekamp–Massey–Sakata Algorithms. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_24
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