M. Return-p-references-alekhnovich, Linear Diophantine equations over polynomials and soft decoding of Reed-Solomon codes, FOCS'02. IEEE, pp.439-448, 2002.

M. Alekhnovich, Linear Diophantine Equations Over Polynomials and Soft Decoding of Reed???Solomon Codes, IEEE Transactions on Information Theory, vol.51, issue.7, pp.2257-2265850097, 2005.
DOI : 10.1109/TIT.2005.850097

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

B. Beckermann, A reliable method for computing M-Pad?? approximants on arbitrary staircases, Journal of Computational and Applied Mathematics, vol.40, issue.1, pp.19-42, 1992.
DOI : 10.1016/0377-0427(92)90039-Z

B. Beckermann and G. 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.804-823, 1994.
DOI : 10.1137/S0895479892230031

B. Beckermann and G. Labahn, Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs, SIAM Journal on Matrix Analysis and Applications, vol.22, issue.1, pp.114-144, 2000.
DOI : 10.1137/S0895479897326912

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

P. Beelen and K. 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

T. G. Beelen, G. J. Van-den-hurk, and C. Praagman, A new method for computing a column reduced polynomial matrix, Systems & Control Letters, vol.10, issue.4, pp.217-2240167, 1988.
DOI : 10.1016/0167-6911(88)90010-2

D. J. Bernstein, Simplified High-Speed High-Distance List Decoding for Alternant Codes, PQCrypto'11, pp.200-216, 2011.
DOI : 10.1016/0022-314X(69)90047-X

A. Bostan, C. Jeannerod, and E. Schost, Solving structured linear systems with large displacement rank, Theor. Comput. Sci, vol.407, pp.1-3, 2008.

K. Brander, Interpolation and list decoding of algebraic codes, 2010.

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.259-2950196, 1980.
DOI : 10.1016/0196-6774(80)90013-9

P. Busse, Multivariate list decoding of evaluation codes with a Gröbner basis perspective, 2008.

D. G. Cantor and E. Kaltofen, On fast multiplication of polynomials over arbitrary algebras, Acta Informatica, vol.7, issue.7, pp.693-701, 1991.
DOI : 10.1007/BF01178683

M. Chowdhury, C. Jeannerod, V. Neiger, E. Schost, and G. Villard, Faster Algorithms for Multivariate Interpolation With Multiplicities and Simultaneous Polynomial Approximations, IEEE Transactions on Information Theory, vol.61, issue.5, pp.2370-2387, 2015.
DOI : 10.1109/TIT.2015.2416068
URL : https://hal.archives-ouvertes.fr/hal-00941435

H. Cohn and N. Heninger, Approximate common divisors via lattices, Tenth Algorithmic Number Theory Symposium, pp.2012-2013
DOI : 10.2140/obs.2013.1.271

H. Cohn and N. 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

D. Coppersmith and S. Winograd, Matrix multiplication via arithmetic progressions, Proceedings of the nineteenth annual ACM conference on Theory of computing , STOC '87, pp.251-280, 1990.
DOI : 10.1145/28395.28396

C. Devet, I. Goldberg, and N. Heninger, Optimally robust private information retrieval, USENIX Security 12. USENIX, pp.269-283, 2012.

D. S. Dummit, R. M. Foote, J. Zur-gathen, and J. Gerhard, Abstract Algebra, Modern Computer Algebra, 2004.

P. 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

S. Gupta, S. Sarkar, A. Storjohann, and J. Valeriote, Triangular <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>x</mml:mi></mml:math>-basis decompositions and derandomization of linear algebra algorithms over <mml:math altimg="si2.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mstyle mathvariant="sans-serif"><mml:mi>K</mml:mi></mml:mstyle><mml:mrow><mml:mo>[</mml:mo><mml:mi>x</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math>, Journal of Symbolic Computation, vol.47, issue.4, pp.422-453, 2012.
DOI : 10.1016/j.jsc.2011.09.006

V. Guruswami and A. 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

V. Guruswami and M. 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

V. Guruswami and M. 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

T. Kailath, Linear Systems, 1980.

W. Keller-gehrig, Fast algorithms for the characteristics polynomial, Theoretical Computer Science, vol.36, pp.309-317, 1985.
DOI : 10.1016/0304-3975(85)90049-0

D. E. Knuth, The analysis of algorithms, In: Congrès int. Math, vol.3, pp.269-274, 1970.

R. Kötter, Fast generalized minimum-distance decoding of algebraic-geometry and Reed-Solomon codes, IEEE Transactions on Information Theory, vol.42, issue.3, pp.721-737, 1996.
DOI : 10.1109/18.490540

R. Kötter and A. Vardy, Algebraic soft-decision decoding of reed-solomon codes, IEEE Transactions on Information Theory, vol.49, issue.11, pp.2809-2825, 2003.
DOI : 10.1109/TIT.2003.819332

R. Kötter and A. Vardy, A complexity reducing transformation in algebraic list decoding of Reed-Solomon codes, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674), pp.10-13, 2003.
DOI : 10.1109/ITW.2003.1216682

L. Gall and F. , 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

K. Lee and M. O-'sullivan, An interpolation algorithm using Gröbner bases for soft-decision decoding of Reed-Solomon codes, IEEE International Symposium on Information Theory, pp.2032-2036, 2006.

K. Lee and M. E. 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. Lübbe, ¨ Uber ein allgemeines Interpolationsproblem ? lineare Identitäten zwischen benachbarten Lösungssystemen, 1983.

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

R. J. Mceliece, The Guruswami-Sudan decoding algorithm for Reed-Solomon codes, IPN Progress Report, pp.42-153, 2003.

T. Mulders and A. Storjohann, On lattice reduction for polynomial matrices, Journal of Symbolic Computation, vol.35, issue.4, pp.377-401, 2003.
DOI : 10.1016/S0747-7171(02)00139-6

J. S. Nielsen, Fast Kötter-Nielsen-Høholdt interpolation in the Guruswami- Sudan algorithm, 2014.

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

V. Olshevsky and M. A. 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

F. Parvaresh and A. 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

S. Paszkowski, Recurrence relations in Padé-Hermite approximation, J. Comput . Appl. Math, vol.19, issue.1, pp.99-107, 1987.

R. M. Roth and G. 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. Schönhage, Schnelle Berechnung von Kettenbruchentwicklungen, Acta Informatica, vol.1, issue.2, pp.139-144, 1971.
DOI : 10.1007/BF00289520

A. V. Sergeyev, A recursive algorithm for Pad??-Hermite approximations, USSR Computational Mathematics and Mathematical Physics, vol.26, issue.2, pp.17-22, 1987.
DOI : 10.1016/0041-5553(86)90003-0

A. Storjohann, Algorithms for matrix canonical forms, 2000.

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

M. Sudan, Decoding of Reed Solomon Codes beyond the Error-Correction Bound, Journal of Complexity, vol.13, issue.1, pp.180-193, 1997.
DOI : 10.1006/jcom.1997.0439

V. Barel, M. Bultheel, and A. , The computation of non-perfect Pad??-Hermite approximants, Numerical Algorithms, vol.33, issue.3, pp.285-304, 1991.
DOI : 10.1007/BF02142327

V. Barel, M. Bultheel, and A. , A general module theoretic framework for vector M-Pad?? and matrix rational interpolation, Numerical Algorithms, vol.1, issue.1, pp.451-462, 1992.
DOI : 10.1007/BF02141952

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

A. Zeh, Algebraic Soft-and Hard-Decision Decoding of Generalized Reed?Solomon and Cyclic Codes, 2013.
URL : https://hal.archives-ouvertes.fr/pastel-00866134

A. Zeh, C. Gentner, and D. Augot, An Interpolation Procedure for List Decoding Reed&#x2013;Solomon Codes Based on Generalized Key Equations, IEEE Transactions on Information Theory, vol.57, issue.9, pp.5946-5959, 2011.
DOI : 10.1109/TIT.2011.2162160

W. Zhou, Fast order basis and kernel basis computation and related problems, 2012.

W. Zhou and G. Labahn, Efficient algorithms for order basis computation, Journal of Symbolic Computation, vol.47, issue.7, pp.793-819, 2012.
DOI : 10.1016/j.jsc.2011.12.009

W. Zhou, G. Labahn, and A. Storjohann, Computing minimal nullspace bases, Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, ISSAC '12, pp.366-373, 2012.
DOI : 10.1145/2442829.2442881