H. Mm-(-· and ·. , 155 MM : multiplication time, 245 MM: mult. time over K[X] m×m , 155 M: multiplication time over K[X], 155 MM : multiplication time, 245 MM : multiplication time, 245 c j : coordinate vector , a m ): diagonal matrix, I m : identity m × m matrix, lm s (A): s-leading matrix, 24 A * ,j : column j of a matrix, A i, * : row i of a matrix, rank(A): rank of a matrix A, A T : transpose of a matrix A, rep(x, J): repetition of x in J, 270 L: leading terms of Gröbner basis, 52 E = {? 1 , . . . , ? D }: monomial basis, 45 B: border (of monomial basis), 52 S: set of exponents, G: Gröbner basis, · · ··: ideal specified by generators, in (M): -initial module, 44 in (f ): -initial term of f , 44 µ: often the multiplicity of a root, pp.41-42

]. M. [-ale02 and . Alekhnovich, Linear Diophantine equations over polynomials and soft decoding of Reed-Solomon codes, FOCS'02, pp.439-448, 2002.

]. M. [-ale05 and . Alekhnovich, Linear Diophantine equations over polynomials and soft decoding of Reed-Solomon codes, IEEE Trans. Inf. Theory, vol.51, issue.7, pp.2257-2265, 2005.

B. [. Bitmead and . Anderson, Asymptotically fast solution of toeplitz and related systems of linear equations, Linear Algebra and its Applications, vol.34, pp.103-11610, 1980.
DOI : 10.1016/0024-3795(80)90161-5

]. G. Bak75 and . Baker, Essentials of Padé Approximants, 1975.

K. [. Beelen and . Brander, Key equations for list decoding of Reed???Solomon codes and how to solve them, Journal of Symbolic Computation, vol.45, issue.7, pp.773-786, 2010.
DOI : 10.1016/j.jsc.2010.03.010

B. [. Berthomieu, J. Boyer, and . Faugère, Linear algebra for computing gröbner bases of linear recursive multidimensional sequences, ISSAC'15, pp.61-68

B. [. Berthomieu, J. Boyer, and . Faugère, Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences, J. Symbolic Comput

]. B. Bec90 and . Beckermann, Zur Interpolation mit polynomialen Linearkombinationen beliebiger Funktionen, 1990.

]. B. Bec92 and . Beckermann, A reliable method for computing M-Padé approximants on arbitrary staircases, J. Comput. Appl. Math, vol.40, issue.192, pp.19-4210, 1992.

]. E. Ber68 and . Berlekamp, Algebraic Coding Theory -Revised edition, 1968.

]. D. Ber11 and . Bernstein, Simplified high-speed high-distance list decoding for alternant codes, PQCrypto'11, pp.200-216, 2011.

J. [. Berthomieu and . Faugère, Guessing linear recurrence relations of sequence tuples and p-recursive sequences with linear algebra, ISSAC'16, pp.95-102
URL : https://hal.archives-ouvertes.fr/hal-01314266

P. [. Baker and . Graves-morris, Padé Approximants. Encyclopedia of Mathematics and its Applications, 1996.

R. P. Brent, F. G. Gustavson, and D. Y. Yun, Fast solution of toeplitz systems of equations and computation of Pad?? approximants, Journal of Algorithms, vol.1, issue.3, pp.259-29510, 1980.
DOI : 10.1016/0196-6774(80)90013-9

J. [. Bunch and . Hopcroft, Triangular factorization and inversion by fast matrix multiplication, Mathematics of Computation, vol.28, issue.125, pp.231-23610, 1974.
DOI : 10.1090/S0025-5718-1974-0331751-8

P. Beelen, T. Høholdt, J. S. Nielsen, and Y. Wu, On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes, IEEE Transactions on Information Theory, vol.59, issue.6, pp.3269-3281, 2013.
DOI : 10.1109/TIT.2013.2243800

URL : http://arxiv.org/abs/1211.0122

C. [. Bostan, C. Jeannerod, É. Mouilleron, and . Schost, On matrices with displacement structure: generalized operators and faster algorithms, 2016.

C. [. Bostan, É. Jeannerod, and . Schost, Solving structured linear systems with large displacement rank, Theoretical Computer Science, vol.407, issue.1-3, pp.155-181, 2008.
DOI : 10.1016/j.tcs.2008.05.014

URL : http://doi.org/10.1016/j.tcs.2008.05.014

G. [. Beckermann and . Labahn, A Uniform Approach for the Fast Computation of Matrix-Type Pad?? Approximants, SIAM Journal on Matrix Analysis and Applications, vol.15, issue.3, pp.804-82310, 1994.
DOI : 10.1137/S0895479892230031

G. [. Beckermann and . Labahn, Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs, SIAM Journal on Matrix Analysis and Applications, vol.22, issue.1, pp.114-14410, 2000.
DOI : 10.1137/S0895479897326912

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

G. [. Beckermann, G. Labahn, and . Villard, Shifted normal forms of polynomial matrices, Proceedings of the 1999 international symposium on Symbolic and algebraic computation , ISSAC '99, pp.189-196, 1999.
DOI : 10.1145/309831.309929

B. Beckermann, G. Labahn, and G. Villard, Normal forms for general polynomial matrices, Journal of Symbolic Computation, vol.41, issue.6, pp.708-737, 2006.
DOI : 10.1016/j.jsc.2006.02.001

URL : http://doi.org/10.1016/j.jsc.2006.02.001

D. Boneh, Finding smooth integers in short intervals using CRT decoding

R. [. Basu, M. Pollack, and . Roy, Algorithms in Real Algebraic Geometry, pp.10-1007, 2006.
DOI : 10.1007/978-3-662-05355-3

URL : https://hal.archives-ouvertes.fr/hal-01083587

]. K. Bra10 and . Brander, Interpolation and List Decoding of Algebraic Codes, 2010.

É. [. Bostan and . Schost, Polynomial evaluation and interpolation on special sets of points, Journal of Complexity, vol.21, issue.4, pp.420-446, 2005.
DOI : 10.1016/j.jco.2004.09.009

B. [. Bostan, É. Salvy, and . Schost, Fast Algorithms for Zero-Dimensional Polynomial Systems using Duality, Applicable Algebra in Engineering, Communication and Computing, vol.14, issue.4, pp.239-272, 2003.
DOI : 10.1007/s00200-003-0133-5

URL : https://hal.archives-ouvertes.fr/inria-00072296

]. B. Buc76 and . Buchberger, A theoretical basis for the reduction of polynomials to canonical forms, ACM SIGSAM Bull, vol.10, issue.3, pp.19-29, 1976.

]. P. Bus08 and . Busse, Multivariate List Decoding of Evaluation Codes with a Gröbner Basis Perspective, 2008.

]. Bvdhp88, . G. Th, G. J. Beelen, C. Van-den-hurk, and . Praagman, A new method for computing a column reduced polynomial matrix, Systems and Control Letters, vol.1088, issue.4, pp.217-22410, 1988.

]. A. Cau21 and . Cauchy, Cours d'analyse de l'École Royale Polytechnique (Analyse algébrique ) ? Sur la formule de Lagrange relative à l'interpolation. Imprimerie royale, p.1821

D. [. Cabay and . Choi, Algebraic Computations of Scaled Pad?? Fractions, SIAM Journal on Computing, vol.15, issue.1
DOI : 10.1137/0215018

N. [. Cohn and . Heninger, Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding, Innovations in Computer Science, pp.298-308, 2011.
DOI : 10.3934/amc.2015.9.311

N. [. Cohn and . Heninger, Approximate common divisors via lattices, Tenth Algorithmic Number Theory Symposium, pp.271-293
DOI : 10.2140/obs.2013.1.271

URL : http://arxiv.org/abs/1108.2714

N. [. Cohn and . Heninger, Ideal forms of Coppersmith's theorem and Guruswami-Sudan list decoding, Advances in Mathematics of Communications, vol.9, issue.3, pp.311-339, 2015.
DOI : 10.3934/amc.2015.9.311

URL : http://arxiv.org/abs/1008.1284

]. U. Che84 and . Cheng, On the continued fraction and Berlekamp's algorithm (corresp.)

. Cjn-+-15-]-m, C. Chowdhury, V. Jeannerod, É. Neiger, G. Schost et al., Faster algorithms for multivariate interpolation with multiplicities and simultaneous polynomial approximations, IEEE Trans. Inf. Theory, vol.61, issue.5, pp.2370-2387, 2015.

E. [. Cantor and . Kaltofen, On fast multiplication of polynomials over arbitrary algebras, Acta Informatica, vol.7, issue.7, pp.693-70110, 1991.
DOI : 10.1007/BF01178683

]. G. Cla75 and . Claessens, A new look at the Padé table and the different methods for computing its elements, J. Comput. Appl. Math, vol.1, issue.375, pp.141-152, 1975.

]. J. Coa66 and . Coates, On the algebraic approximation of functions, I?III, Indag. Math, vol.69, issue.66, pp.421-46110, 1966.

]. J. Coa67 and . Coates, On the algebraic approximation of functions, IV. Indag. Math, vol.70, issue.67, pp.205-21210, 1967.

[. Coppersmith, Finding a Small Root of a Univariate Modular Equation, Lecture Notes in Computer Science, vol.1070, pp.155-16510, 1996.
DOI : 10.1007/3-540-68339-9_14

S. [. Coppersmith and . Winograd, Matrix multiplication via arithmetic progressions, Proceedings of the nineteenth annual ACM conference on Theory of computing , STOC '87, pp.251-28010, 1990.
DOI : 10.1145/28395.28396

URL : http://doi.org/10.1016/s0747-7171(08)80013-2

]. A. Dan37 and . Danilevskii, The numerical solution of the secular equation, Matem. Sbornik, vol.44, issue.2, pp.169-171, 1937.

R. [. Dummit and . Foote, Abstract Algebra, 2004.

C. Devet, I. Goldberg, and N. Heninger, Optimally robust private information retrieval, USENIX Security 12, pp.269-283

R. [. Demillo and . Lipton, A probabilistic remark on algebraic program testing, Information Processing Letters, vol.7, issue.4, pp.193-19590067, 1978.
DOI : 10.1016/0020-0190(78)90067-4

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

]. J. Dor87 and . Dornstetter, On the equivalence between Berlekamp's and Euclid's algorithms, IEEE Trans. Inf. Theory, vol.33, issue.3, pp.428-431, 1987.

W. Eberly, M. Giesbrecht, and G. Villard, On computing the determinant and Smith form of an integer matrix, Proceedings 41st Annual Symposium on Foundations of Computer Science, pp.675-687, 2000.
DOI : 10.1109/SFCS.2000.892335

]. D. Eis95 and . Eisenbud, Commutative Algebra: with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, pp.10-1007, 1995.

]. D. Eis05 and . Eisenbud, The Geometry of Syzygies. Graduate Texts in Mathematics, pp.10-1007, 2005.

J. [. Fitzpatrick and . Flynn, A Gr??bner basis technique for Pad?? approximation, Journal of Symbolic Computation, vol.13, issue.2, pp.133-13810, 1992.
DOI : 10.1016/S0747-7171(08)80087-9

S. [. Farr and . Gao, Gröbner bases and generalized Padé approximation, Mathematics of Computation, vol.74, issue.1, pp.461-473, 2006.

[. Faugère, P. Gaudry, L. Huot, and G. Renault, Polynomial systems solving by fast linear algebra

[. Faugère, P. Gaudry, L. Huot, and G. Renault, Sub-cubic change of ordering for Gröbner basis: a probabilistic approach, ISSAC'14, pp.170-177

J. Faugère, P. Gianni, D. Lazard, and T. Mora, Efficient Computation of Zero-dimensional Gr??bner Bases by Change of Ordering, Journal of Symbolic Computation, vol.16, issue.4, pp.329-344, 1993.
DOI : 10.1006/jsco.1993.1051

]. P. Fit95 and . Fitzpatrick, On the key equation, IEEE Trans. Inf. Theory, vol.41, issue.5, pp.1290-1302, 1995.

]. P. Fit97 and . Fitzpatrick, Solving a Multivariable Congruence by Change of Term Order

[. Faugère and C. Mou, Fast algorithm for change of ordering of zerodimensional gröbner bases with sparse multiplication matrices, ISSAC'11, pp.115-12210, 2011.

[. Faugère and C. Mou, Sparse FGLM algorithms, Journal of Symbolic Computation, vol.80, issue.3, pp.538-569
DOI : 10.1016/j.jsc.2016.07.025

]. G. For75 and J. Forney, Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems, SIAM Journal on Control, vol.13, issue.3, pp.493-52010, 1975.

K. [. Feng and . Tzeng, A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes, IEEE Transactions on Information Theory, vol.37, issue.5, pp.1274-1287, 1991.
DOI : 10.1109/18.133246

]. K. Ged73 and . Geddes, Algorithms for Analytic Approximation (to a Formal Powerseries ), Canada, 1973.

]. K. Ged79 and . Geddes, Symbolic computation of Padé approximants, ACM Trans. Math. Softw, vol.5, issue.2, pp.218-233, 1979.

[. Giorgi, C. Jeannerod, and G. Villard, On the complexity of polynomial matrix computations, Proceedings of the 2003 international symposium on Symbolic and algebraic computation , ISSAC '03, pp.135-142, 2003.
DOI : 10.1145/860854.860889

[. Giorgi and R. Lebreton, Online order basis algorithm and its impact on the block Wiedemann algorithm, Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ISSAC '14, pp.202-209
DOI : 10.1145/2608628.2608647

URL : https://hal.archives-ouvertes.fr/lirmm-01232873

O. [. Gaborit and . Ruatta, Improved Hermite multivariate polynomial interpolation, 2006 IEEE International Symposium on Information Theory, pp.143-147, 2006.
DOI : 10.1109/ISIT.2006.261691

A. [. Guruswami and . Rudra, Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy, IEEE Transactions on Information Theory, vol.54, issue.1, pp.135-150, 2008.
DOI : 10.1109/TIT.2007.911222

URL : http://arxiv.org/abs/cs/0511072

M. [. Guruswami and . Sudan, Improved decoding of Reed-Solomon and algebraic-geometric codes, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), pp.28-39, 1998.
DOI : 10.1109/SFCS.1998.743426

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

M. [. Guruswami and . Sudan, Improved decoding of Reed-Solomon and algebraic-geometry codes, IEEE Transactions on Information Theory, vol.45, issue.6, pp.1757-1767, 1999.
DOI : 10.1109/18.782097

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

S. Gupta and A. Storjohann, Computing hermite forms of polynomial matrices, Proceedings of the 36th international symposium on Symbolic and algebraic computation, ISSAC '11, pp.155-162, 2011.
DOI : 10.1145/1993886.1993913

S. [. Gupta, A. Sarkar, J. Storjohann, and . Valeriote, Triangular x-basis decompositions and derandomization of linear algebra algorithms

M. Girault, P. Toffin, and B. Vallée, Computation of Approximate L-th Roots Modulo n and Application to Cryptography, CRYPTO '88, pp.100-11710, 1990.
DOI : 10.1007/0-387-34799-2_9

]. S. Gup11 and . Gupta, Hermite forms of polynomial matrices, 2011.

D. [. Gustavson and . Yun, Fast algorithms for rational Hermite approximation and solution of Toeplitz systems, IEEE Transactions on Circuits and Systems, vol.26, issue.9, pp.750-755, 1979.
DOI : 10.1109/TCS.1979.1084696

]. H. Has36 and . Hasse, Theorie der höheren Differentiale in einem algebraischen Funktionenkörper mit vollkommenem Konstantenkörper bei beliebiger Charakteristik, J. Reine Angew. Math, vol.175, pp.50-54, 1936.

J. Håstad, On using RSA with low exponent in a public key network In Lecture Notes in Computer Sciences; 218 on Advances in Cryptology? CRYPTO'85, pp.403-40810, 1986.

]. C. Her51 and . Hermite, Sur l'introduction des variables continues dans la théorie des nombres, Journal für die reine und angewandte Mathematik, vol.41, pp.191-216

]. C. Her93 and . Hermite, Sur la généralisation des fractions continues algébriques, pp.289-308

. [. Howgrave-graham, Approximate Integer Common Divisors, CaLC'01, pp.51-66, 2001.
DOI : 10.1007/3-540-44670-2_6

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

K. [. Hafner and . Mccurley, Asymptotically Fast Triangularization of Matrices over Rings, SIAM Journal on Computing, vol.20, issue.6, pp.1068-108310, 1991.
DOI : 10.1137/0220067

]. C. Ili89 and . Iliopoulos, Worst-case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix, SIAM J. Comput, vol.18, issue.4, pp.658-669, 1989.

S. [. Ibarra, R. Moran, and . Hui, A generalization of the fast LUP matrix decomposition algorithm and applications, Journal of Algorithms, vol.3, issue.1, pp.45-5610, 1982.
DOI : 10.1016/0196-6774(82)90007-4

]. H. Jag64 and . Jager, A multidimensional generalization of the Padé table, I?VI, Indag. Math, vol.6764, pp.193-24910, 1964.

C. Jeannerod, V. Neiger, É. Schost, and G. Villard, Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts, Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC '16, pp.295-302, 2016.
DOI : 10.1145/2930889.2930928

URL : https://hal.archives-ouvertes.fr/hal-01265983

C. Jeannerod, V. Neiger, É. Schost, and G. Villard, Computing minimal interpolation bases, Journal of Symbolic Computation, 2017.
DOI : 10.1016/j.jsc.2016.11.015

URL : https://hal.archives-ouvertes.fr/hal-01241781

[. Jeannerod and G. Villard, Essentially optimal computation of the inverse of generic polynomial matrices, Journal of Complexity, vol.21, issue.1, pp.72-86, 2005.
DOI : 10.1016/j.jco.2004.03.005

]. T. Kai80 and . Kailath, Linear Systems, 1980.

]. E. Kal94 and . Kaltofen, Asymptotically fast solution of Toeplitz-like singular linear systems, ISSAC'94, pp.297-30410, 1994.

. [. Keller-gehrig, Fast algorithms for the characteristics polynomial, Theoretical Computer Science, vol.36, pp.309-31710, 1985.
DOI : 10.1016/0304-3975(85)90049-0

URL : http://doi.org/10.1016/0304-3975(85)90049-0

M. [. Kaltofen, D. Krishnamoorthy, and . Saunders, Parallel algorithms for matrix normal forms, Linear Algebra and its Applications, vol.136, issue.90, pp.189-208, 1990.
DOI : 10.1016/0024-3795(90)90028-B

J. [. Kötter, A. Ma, and . Vardy, The Re-Encoding Transformation in Algebraic List-Decoding of Reed–Solomon Codes, IEEE Transactions on Information Theory, vol.57, issue.2, pp.633-647, 2011.
DOI : 10.1109/TIT.2010.2096034

]. D. Knu70 and . Knuth, The analysis of algorithms, Congrès int, pp.269-274, 1970.

]. R. Köt96 and . Kötter, Fast generalized minimum-distance decoding of algebraic-geometry and Reed-Solomon codes, IEEE Trans. Inf. Theory, vol.42, issue.3, pp.721-737, 1996.

]. A. Kry31 and . Krylov, On the numerical solution of the equation by which, in technical questions, frequencies of small oscillations of material systems are determined, Izvestiya Akademii Nauk SSSR, vol.7, issue.4, pp.491-539, 1931.

D. [. Kaltofen and . Saunders, On wiedemann's method of solving sparse linear systems, AAECC-9, pp.29-38, 1991.
DOI : 10.1007/3-540-54522-0_93

C. [. Kedlaya and . Umans, Fast Polynomial Factorization and Modular Composition, SIAM Journal on Computing, vol.40, issue.6, pp.1767-1802, 2011.
DOI : 10.1137/08073408X

URL : http://authors.library.caltech.edu/28972/1/Kedlaya2011p16787Siam_J_Comput.pdf

]. R. Kv03a, A. Kötter, and . Vardy, Algebraic soft-decision decoding of Reed-Solomon codes, IEEE Trans. Inf. Theory, issue.11, pp.492809-2825, 2003.

]. R. Kv03b, A. Kötter, and . Vardy, A complexity reducing transformation in algebraic list decoding of Reed-Solomon codes, ITW2003, pp.10-13, 2003.

[. Gall, Powers of tensors and fast matrix multiplication, Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ISSAC '14, pp.296-303, 2014.
DOI : 10.1145/2608628.2608664

V. [. Labahn, W. Neiger, and . Zhou, Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01345627

M. [. Lee and . O-'sullivan, An interpolation algorithm using Gröbner bases for soft-decision decoding of Reed-Solomon codes, ISIT'06, pp.2032-2036, 2006.

M. [. Lee and . O-'sullivan, List decoding of Reed???Solomon codes from a Gr??bner basis perspective, Journal of Symbolic Computation, vol.43, issue.9, pp.645-658, 2008.
DOI : 10.1016/j.jsc.2008.01.002

]. W. Lüb83 and . Lübbe, Über ein allgemeines Interpolationsproblem ? lineare Identitäten zwischen benachbarten Lösungssystemen, 1983.

]. F. Mac02 and . Macaulay, Some formulae in elimination, Proceedings of the London Mathematical Society, pp.1-353, 1902.

]. F. Mac16 and . Macaulay, The Algebraic Theory of Modular Systems. Cambridge Tracts in Mathematics and Mathematical Physics, 1916.

]. K. Mah32 and . Mahler, Zur approximation der Exponentialfunktion und des Logarithmus

]. K. Mah53 and . Mahler, On the approximation of logarithms of algebraic numbers, Philosophical Transactions of the Royal Society of London, Series A, vol.245, issue.898, pp.371-398, 1953.

]. K. Mah68 and . Mahler, Perfect systems, Composit. Math, vol.19, issue.2, pp.95-166, 1968.

J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Transactions on Information Theory, vol.15, issue.1
DOI : 10.1109/TIT.1969.1054260

B. [. Möller and . Buchberger, The construction of multivariate polynomials with preassigned zeros, EUROCAM'82, pp.24-31, 1982.
DOI : 10.1007/3-540-11607-9_3

]. R. Mce03 and . Mceliece, The Guruswami-Sudan decoding algorithm for Reed-Solomon codes, 2003.

]. J. Mid11 and . Middeke, A computational view on normal forms of matrices of Ore polynomials, Risc technical report 11-10, Research Institute for Symbolic Computation (RISC), 2011.

M. G. Marinari, H. M. Möller, and T. Mora, Gr???bner bases of ideals defined by functionals with an application to ideals of projective points, Applicable Algebra in Engineering, Communication and Computing, vol.100, issue.2, pp.103-14510, 1993.
DOI : 10.1007/BF01386834

]. R. Moe73 and . Moenck, Fast computation of GCDs, Proc. 5th ACM Symp. Theory Comp, pp.142-151, 1973.

]. M. Mor80 and . Morf, Doubling algorithms for Toeplitz and related equations, IEEE Conference on Acoustics, Speech, and Signal Processing, pp.954-959

]. T. Mor09, M. Mora, S. Sala, T. Sakata, C. Mora et al., The FGLM problem and Möller's algorithm on zero-dimensional ideals, Gröbner Bases, Coding, and Cryptography, pp.27-4510, 2009.

R. J. Mceliece and J. B. Shearer, A Property of Euclid???s Algorithm and an Application to Pad?? Approximation, SIAM Journal on Applied Mathematics, vol.34, issue.4, pp.611-61510, 1978.
DOI : 10.1137/0134048

. [. Moreno-socias, Autour de la fonction de Hilbert-Samuel (escaliers d'idéaux polynomiaux), 1991.

]. G. Ms03a and . Moreno-socias, Degrevlex Gröbner bases of generic complete intersections, Journal of Pure and Applied Algebra, vol.180, issue.302, pp.263-28310, 2003.

]. T. Ms03b, A. Mulders, and . Storjohann, On lattice reduction for polynomial matrices, J. Symbolic Comput, vol.35, issue.02, pp.377-40110, 2003.

B. [. Miller and . Sturmfels, Combinatorial Commutative Algebra. Graduate texts in mathematics, 2005.

]. V. Nei16 and . Neiger, Fast computation of shifted Popov forms of polynomial matrices via systems of modular polynomial equations, ISSAC'16, pp.365-372

T. [. Nielsen and . Høholdt, Decoding Reed-Solomon Codes Beyond Half the Minimum Distance, Coding Theory, Cryptography and Related Areas, pp.221-236, 2000.
DOI : 10.1007/978-3-642-57189-3_20

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

]. J. Nie13 and . Nielsen, List Decoding of Algebraic Codes, 2013.

]. J. Nie14 and . Nielsen, Fast Kötter-Nielsen-Høholdt interpolation in the Guruswami- Sudan algorithm, ACCT'14, 2014.

]. H. Nus80 and . Nussbaumer, Fast polynomial transform algorithms for digital convolution, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.28, issue.2, pp.205-215, 1980.

M. [. Nüsken and . Ziegler, Fast Multipoint Evaluation of Bivariate Polynomials, Algorithms ? ESA 2004, pp.544-555, 2004.
DOI : 10.1007/978-3-540-30140-0_49

P. [. O-'keeffe and . Fitzpatrick, Gröbner basis solutions of constrained interpolation problems, Linear Algebra Appl, vol.351, issue.01, pp.533-551, 2002.

P. [. O-'keeffe and . Fitzpatrick, Gröbner basis approach to list decoding of algebraic geometry codes, Appl. Algebra Engrg. Comm. Comput, vol.18, issue.5, pp.445-466, 2007.

M. [. Olshevsky and . Shokrollahi, A displacement approach to efficient decoding of algebraic-geometric codes, Proceedings of the thirty-first annual ACM symposium on Theory of computing , STOC '99, pp.235-244, 1999.
DOI : 10.1145/301250.301311

]. H. Pad94 and . Padé, Sur la généralisation des fractions continues algébriques, Journal de Mathématiques Pures et Appliquées, pp.291-330, 1894.

]. C. Pap94 and . Papadimitriou, Computational Complexity. Theoretical Computer Science, 1994.

]. S. Pas87 and . Paszkowski, Recurrence relations in Padé-Hermite approximation, J. Comput . Appl. Math, vol.1987, issue.1, pp.99-10710, 1987.

]. V. Pop72 and . Popov, Invariant description of linear, time-invariant controllable systems, SIAM Journal on Control, vol.10, issue.2, pp.252-264, 1972.

A. [. Parvaresh and . Vardy, Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pp.285-294, 2005.
DOI : 10.1109/SFCS.2005.29

]. [-rei03 and . Reinhard, Algorithme LLL polynomial et applications Master's thesis, École Polytechnique URL: https, 2003.

]. L. Rob86 and . Robbiano, On the theory of graded structures, J. Symbolic Comput, vol.2, issue.286, pp.139-17010, 1986.

]. R. Rot07 and . Roth, Introduction to Coding Theory, 2007.

]. F. Rou99 and . Rouillier, Solving zero-dimensional systems through the rational univariate representation, Appl. Algebra Engrg. Comm. Comput, vol.9, issue.5, pp.433-46110, 1999.

G. [. Roth and . Ruckenstein, Efficient decoding of Reed-Solomon codes beyond half the minimum distance, IEEE Transactions on Information Theory, vol.46, issue.1, pp.246-257, 2000.
DOI : 10.1109/18.817522

A. [. Rosenkilde and . Storjohann, Algorithms for Simultaneous Pad?? Approximations, Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC '16, pp.405-412
DOI : 10.1145/2930889.2930933

]. S. Sak88 and . Sakata, Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array, J. Symbolic Comput, vol.588, issue.3, pp.321-33710, 1988.

S. Sakata, Extension of the Berlekamp-Massey algorithm to N dimensions, Information and Computation, vol.84, issue.2, pp.207-23910, 1990.
DOI : 10.1016/0890-5401(90)90039-K

]. J. Sch80 and . Schwartz, Fast probabilistic algorithms for verification of polynomial identities, J. ACM, vol.27, issue.4, pp.701-717, 1980.

]. A. Ser87 and . Sergeyev, A recursive algorithm for Padé-Hermite approximations

]. V. Sho91 and . Shoup, A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic, ISSAC'91, pp.14-21, 1991.

M. [. Sugiyama, S. Kasahara, T. Hirasawa, and . Namekawa, A method for solving key equation for decoding goppa codes, Information and Control, vol.27, issue.1, pp.87-9910, 1975.
DOI : 10.1016/S0019-9958(75)90090-X

URL : http://doi.org/10.1016/s0019-9958(75)90090-x

G. [. Storjohann and . Labahn, Asymptotically fast computation of Hermite normal forms of integer matrices, Proceedings of the 1996 international symposium on Symbolic and algebraic computation , ISSAC '96, pp.259-266, 1996.
DOI : 10.1145/236869.237083

A. [. Sarkar and . Storjohann, Normalization of row reduced matrices, Proceedings of the 36th international symposium on Symbolic and algebraic computation, ISSAC '11, pp.297-30410, 2011.
DOI : 10.1145/1993886.1993931

]. A. Sto00 and . Storjohann, Algorithms for Matrix Canonical Forms, 2000.

]. A. Sto03 and . Storjohann, High-order lifting and integrality certification, J. Symbolic Comput, vol.36, issue.3-403, pp.613-64810, 2003.

]. A. Sto06 and . Storjohann, Notes on computing minimal approximant bases, Challenges in Symbolic Computation Software, Dagstuhl Seminar Proceedings, p.776, 2006.

]. M. Sud97 and . Sudan, Decoding of Reed-Solomon codes beyond the error-correction bound, J. Complexity, vol.13, issue.1, pp.180-193, 1997.

G. [. Storjohann and . Villard, Computing the rank and a small nullspace basis of a polynomial matrix, Proceedings of the 2005 international symposium on Symbolic and algebraic computation , ISSAC '05, pp.309-316, 2005.
DOI : 10.1145/1073884.1073927

URL : https://hal.archives-ouvertes.fr/hal-00004832

]. J. Syl53 and . Sylvester, On a Theory of the Syzygetic Relations of Two Rational Integral Functions, Comprising an Application to the Theory of Sturm's Functions, and That of the Greatest Algebraical Common Measure, Philosophical Transactions of the Royal Society of London, vol.143, pp.407-548

]. E. Tho01 and . Thomé, Fast computation of linear generators for matrix sequences and application to the block Wiedemann algorithm, ISSAC'01, pp.323-331, 2001.

]. E. Tho02 and . Thomé, Subquadratic computation of vector generating polynomials and improvement of the block Wiedemann algorithm, J. Symbolic Comput, vol.33, issue.5, pp.757-775, 2002.

]. P. Tri10 and . Trifonov, Efficient interpolation in the Guruswami-Sudan algorithm

. M. Vbb91, . Van, A. Barel, and . Bultheel, The computation of non-perfect Padé- Hermite approximants, Numer. Algorithms, vol.1, issue.3, pp.285-30410, 1991.

. M. Vbb92, . Van, A. Barel, and . Bultheel, A general module theoretic framework for vector M-Padé and matrix rational interpolation, Numer. Algorithms, vol.3, pp.451-46210, 1992.

]. G. Vil96 and . Villard, Computing Popov and Hermite forms of polynomial matrices, ISSAC'96, pp.250-258, 1996.

]. D. War74 and . Warner, Hermite interpolation with rational functions, 1974.

]. D. War76 and . Warner, Kronecker's algorithm for Hermite interpolation with an application to sphere drag in a fluid-filled tube, Proceedings of a Workshop on Padé Approximation, pp.48-71, 1976.

E. [. Welch and . Berlekamp, Error correction for algebraic block codes, US Patent, vol.4633, p.470, 1986.

R. [. Welch and . Scholtz, Continued fractions and Berlekamp's algorithm, IEEE Transactions on Information Theory, vol.25, issue.1
DOI : 10.1109/TIT.1979.1055987

Y. Wu, New List Decoding Algorithms for Reed–Solomon and BCH Codes, IEEE Transactions on Information Theory, vol.54, issue.8, pp.3611-3630, 2008.
DOI : 10.1109/TIT.2008.926355

]. A. Zeh13 and . Zeh, Algebraic Soft-and Hard-Decision Decoding of Generalized Reed? Solomon and Cyclic Codes, 2013.

C. [. Zeh, D. Gentner, and . Augot, An Interpolation Procedure for List Decoding Reed–Solomon Codes Based on Generalized Key Equations, IEEE Transactions on Information Theory, vol.57, issue.9
DOI : 10.1109/TIT.2011.2162160

]. W. Zho12 and . Zhou, Fast Order Basis and Kernel Basis Computation and Related Problems, 2012.

]. N. Zie68 and . Zierler, Linear recurring sequences and error-correcting codes

]. R. Zip79 and . Zippel, Probabilistic algorithms for sparse polynomials, EUROSAM'79, pp.216-226, 1979.

G. [. Zhou and . Labahn, Efficient algorithms for order basis computation, Journal of Symbolic Computation, vol.47, issue.7, pp.793-819
DOI : 10.1016/j.jsc.2011.12.009

URL : http://dx.doi.org/10.1016/j.jsc.2011.12.009

G. [. Zhou and . Labahn, Computing column bases of polynomial matrices, Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation, ISSAC '13, pp.379-386
DOI : 10.1145/2465506.2465947

]. W. Zl14a, G. Zhou, and . Labahn, Fast and deterministic computation of the determinant of a polynomial matrix, 2014.

]. W. Zl14b, G. Zhou, and . Labahn, Unimodular completion of polynomial matrices, pp.413-420

G. [. Zhou and . Labahn, A fast, deterministic algorithm for computing a Hermite normal form of a polynomial matrix, 2016.

G. [. Zhou, A. Labahn, and . Storjohann, Computing minimal nullspace bases, Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, ISSAC '12, pp.366-373
DOI : 10.1145/2442829.2442881

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=

G. [. Zhou, A. Labahn, and . Storjohann, A deterministic algorithm for inverting a polynomial matrix, Journal of Complexity, vol.31, issue.2, pp.162-173, 2015.
DOI : 10.1016/j.jco.2014.09.004