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Syndrome decoding for Hermite codes with a Sugiyama-type algorithm
1. | Institute of Pure Mathematics, Ulm University, Ulm, Germany |
2. | Institute of Communications Engineering, Ulm University, Ulm, Germany |
References:
[1] |
P. Beelen and T. Høholdt, The decoding of algebraic geometry codes, in "Advances in algebraic geometry codes, Ser. Coding Theory Cryptol.,'' 5 (2008), 99-152. |
[2] |
D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry,'' Springer, 1998. |
[3] |
I. M. Duursma, Algebraic geometry codes: general theory, in "Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol.,'' 5 (2008), 99-152. |
[4] |
D. Ehrhard, "Über das Dekodieren algebraisch-geometrischer Codes,'' Dissertation, Universität Düsseldorf, 1991. |
[5] |
J. I. Farrán, Decoding algebraic geometry codes by the key equation, Finite Field Appl., 6 (2000), 207-217.
doi: 10.1006/ffta.1999.0274. |
[6] |
G. L. Feng and T. R. N. Rao, Decoding algebraic-geometric codes up to the designed distance, IEEE Trans. Inform. Theory, 39 (1993), 37-45.
doi: 10.1109/18.179340. |
[7] |
V. D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR, 259 (1981), 1289-1290. |
[8] |
V. Guruswami and M. Sudan, Improved decoding of Reed-Solomon and algebraic-geometric codes, in "39th Annual Symposium on Foundations of Computer Science,'' 1998. |
[9] |
T. Høholdt, J. H. van Lint and R. Pellikaan, On the decoding of algebraic-geometric codes, IEEE Trans. Inform. Theory, 41 (1995), 1589-1614.
doi: 10.1109/18.476214. |
[10] |
T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes, in "Handbook of Coding Theory,'' 1 (1998), 871-961. |
[11] |
J. Justesen, K. Larsen, H. Jensen and T. Høholdt, Fast decoding of codes from algebraic plane curves, IEEE Trans. Inform. Theory, 38 (1992), 111-119.
doi: 10.1109/18.108255. |
[12] |
S. Kampf, Bounds on collaborative decoding of interleaved Hermitian codes with a division algorithm and virtual extension, 3ICMCTA Special Issue of Designs, Codes and Cryptography, accepted, 2012. |
[13] |
S. Kampf, M. Bossert and S. Bezzateev, Some results on list decoding of interleaved Reed-Solomon codes with the extended euclidean elgorithm, in "Proc. Coding Theory Days in St. Petersburg 2008,'' (2008), 31-36. |
[14] |
S. Kampf, M. Bossert and I. I. Bouw, Solving the key equation for Hermitian codes with a division algorithm, in "IEEE International Symposium on Information Theory,'' St. Petersburg, (2011), 1008-1012. |
[15] |
D. A. Leonard, A generalized Forney formula for algebraic-geometric codes, IEEE Trans. Inform. Theory, 42 (1996), 1263-1268.
doi: 10.1109/18.508855. |
[16] |
F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland Mathematical Library, 1988. |
[17] |
M. O'Sullivan and M. Bras-Amorós, The key equation for one-point codes, in "Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol.,'' 5 (2008), 99-152. |
[18] |
S. C. Porter, B.-Z. Shen and R. Pellikaan, Decoding geometric Goppa codes using an extra place, IEEE Trans. Inform. Theory, 38 (1992), 1663-1676.
doi: 10.1109/18.165441. |
[19] |
R. M. Roth, "Introduction to Coding Theory,'' Cambridge University Press, 2006. |
[20] |
S. Sakata, J. Justesen, Y. Madelung, H. Elbrønd and T. Høholdt, Fast decoding of algebraic-geometric codes up to the designed minimum distance, IEEE Trans. Inform. Theory, 41 (1995), 1672-1677.
doi: 10.1109/18.476240. |
[21] |
B.-Z. Shen, Solving a congruence on a graded algebra by a subresultant sequence and its application, J. Symbolic Comput., 14 (1992), 505-522.
doi: 10.1016/0747-7171(92)90020-5. |
[22] |
A. Skorobogatov and S. G. Vlăduţ, On the decoding of algebraic-geometric codes, IEEE Trans. Inform. Theory, IT-36 (1990), 1051-1060.
doi: 10.1109/18.57204. |
[23] |
H. Stichenoth, "Algebraic Function Fields and Codes,'' 2nd edition, Springer-Verlag, 2009. |
[24] |
M. A. Tsfasman and S. G. Vlăduţ, "Algebraic-Geometric Codes,'' Kluwer Academic Publishers Group, 1991. |
[25] |
M. A. Tsfasman and S. G. Vlăduţ and T. Zink, Modular curves, Shimura curves and Goppa codes better than the Varshmov-Gilbert bound, Math. Nachr., 109 (1982), 21-28.
doi: 10.1002/mana.19821090103. |
[26] |
K. Yang and P. V. Kumar, On the true minimal distance of Hermitian codes, in "Coding Theory and Algebraic Geometry (Luminy, 1991),'' Springer, Berlin, (1992), 99-107.
doi: 10.1007/BFb0087995. |
show all references
References:
[1] |
P. Beelen and T. Høholdt, The decoding of algebraic geometry codes, in "Advances in algebraic geometry codes, Ser. Coding Theory Cryptol.,'' 5 (2008), 99-152. |
[2] |
D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry,'' Springer, 1998. |
[3] |
I. M. Duursma, Algebraic geometry codes: general theory, in "Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol.,'' 5 (2008), 99-152. |
[4] |
D. Ehrhard, "Über das Dekodieren algebraisch-geometrischer Codes,'' Dissertation, Universität Düsseldorf, 1991. |
[5] |
J. I. Farrán, Decoding algebraic geometry codes by the key equation, Finite Field Appl., 6 (2000), 207-217.
doi: 10.1006/ffta.1999.0274. |
[6] |
G. L. Feng and T. R. N. Rao, Decoding algebraic-geometric codes up to the designed distance, IEEE Trans. Inform. Theory, 39 (1993), 37-45.
doi: 10.1109/18.179340. |
[7] |
V. D. Goppa, Codes on algebraic curves, Dokl. Akad. Nauk SSSR, 259 (1981), 1289-1290. |
[8] |
V. Guruswami and M. Sudan, Improved decoding of Reed-Solomon and algebraic-geometric codes, in "39th Annual Symposium on Foundations of Computer Science,'' 1998. |
[9] |
T. Høholdt, J. H. van Lint and R. Pellikaan, On the decoding of algebraic-geometric codes, IEEE Trans. Inform. Theory, 41 (1995), 1589-1614.
doi: 10.1109/18.476214. |
[10] |
T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes, in "Handbook of Coding Theory,'' 1 (1998), 871-961. |
[11] |
J. Justesen, K. Larsen, H. Jensen and T. Høholdt, Fast decoding of codes from algebraic plane curves, IEEE Trans. Inform. Theory, 38 (1992), 111-119.
doi: 10.1109/18.108255. |
[12] |
S. Kampf, Bounds on collaborative decoding of interleaved Hermitian codes with a division algorithm and virtual extension, 3ICMCTA Special Issue of Designs, Codes and Cryptography, accepted, 2012. |
[13] |
S. Kampf, M. Bossert and S. Bezzateev, Some results on list decoding of interleaved Reed-Solomon codes with the extended euclidean elgorithm, in "Proc. Coding Theory Days in St. Petersburg 2008,'' (2008), 31-36. |
[14] |
S. Kampf, M. Bossert and I. I. Bouw, Solving the key equation for Hermitian codes with a division algorithm, in "IEEE International Symposium on Information Theory,'' St. Petersburg, (2011), 1008-1012. |
[15] |
D. A. Leonard, A generalized Forney formula for algebraic-geometric codes, IEEE Trans. Inform. Theory, 42 (1996), 1263-1268.
doi: 10.1109/18.508855. |
[16] |
F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland Mathematical Library, 1988. |
[17] |
M. O'Sullivan and M. Bras-Amorós, The key equation for one-point codes, in "Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol.,'' 5 (2008), 99-152. |
[18] |
S. C. Porter, B.-Z. Shen and R. Pellikaan, Decoding geometric Goppa codes using an extra place, IEEE Trans. Inform. Theory, 38 (1992), 1663-1676.
doi: 10.1109/18.165441. |
[19] |
R. M. Roth, "Introduction to Coding Theory,'' Cambridge University Press, 2006. |
[20] |
S. Sakata, J. Justesen, Y. Madelung, H. Elbrønd and T. Høholdt, Fast decoding of algebraic-geometric codes up to the designed minimum distance, IEEE Trans. Inform. Theory, 41 (1995), 1672-1677.
doi: 10.1109/18.476240. |
[21] |
B.-Z. Shen, Solving a congruence on a graded algebra by a subresultant sequence and its application, J. Symbolic Comput., 14 (1992), 505-522.
doi: 10.1016/0747-7171(92)90020-5. |
[22] |
A. Skorobogatov and S. G. Vlăduţ, On the decoding of algebraic-geometric codes, IEEE Trans. Inform. Theory, IT-36 (1990), 1051-1060.
doi: 10.1109/18.57204. |
[23] |
H. Stichenoth, "Algebraic Function Fields and Codes,'' 2nd edition, Springer-Verlag, 2009. |
[24] |
M. A. Tsfasman and S. G. Vlăduţ, "Algebraic-Geometric Codes,'' Kluwer Academic Publishers Group, 1991. |
[25] |
M. A. Tsfasman and S. G. Vlăduţ and T. Zink, Modular curves, Shimura curves and Goppa codes better than the Varshmov-Gilbert bound, Math. Nachr., 109 (1982), 21-28.
doi: 10.1002/mana.19821090103. |
[26] |
K. Yang and P. V. Kumar, On the true minimal distance of Hermitian codes, in "Coding Theory and Algebraic Geometry (Luminy, 1991),'' Springer, Berlin, (1992), 99-107.
doi: 10.1007/BFb0087995. |
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