Syndrome decoding for Hermite codes with a Sugiyama-type algorithm

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November 2012, 6(4): 419-442. doi: 10.3934/amc.2012.6.419

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

Received July 2011 Revised May 2012 Published November 2012

Abstract
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This paper gives a new approach to decoding Hermite codes using the key equation, avoiding the use of majority voting. Our approach corrects up to $(d_{\min}-1)/2$ errors, and works up to some extent also beyond. We present an efficient implementation of our algorithm based on a Sugiyama-type iterative procedure for computing solutions of a key equation.

Keywords: Hermite codes, Groebner bases., algebraic decoding.

Mathematics Subject Classification: Primary: 14G50; Secondary: 11T7.

Citation: Irene I. Bouw, Sabine Kampf. Syndrome decoding for Hermite codes with a Sugiyama-type algorithm. Advances in Mathematics of Communications, 2012, 6 (4) : 419-442. doi: 10.3934/amc.2012.6.419

References:

[1]	P. Beelen and T. Høholdt, The decoding of algebraic geometry codes,, in, 5 (2008), 99. Google Scholar
[2]	D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry,'', Springer, (1998). Google Scholar
[3]	I. M. Duursma, Algebraic geometry codes: general theory,, in, 5 (2008), 99. Google Scholar
[4]	D. Ehrhard, "Über das Dekodieren algebraisch-geometrischer Codes,'' Dissertation,, Universität Düsseldorf, (1991). Google Scholar
[5]	J. I. Farrán, Decoding algebraic geometry codes by the key equation,, Finite Field Appl., 6 (2000), 207. doi: 10.1006/ffta.1999.0274. Google Scholar
[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. doi: 10.1109/18.179340. Google Scholar
[7]	V. D. Goppa, Codes on algebraic curves,, Dokl. Akad. Nauk SSSR, 259 (1981), 1289. Google Scholar
[8]	V. Guruswami and M. Sudan, Improved decoding of Reed-Solomon and algebraic-geometric codes,, in, (1998). Google Scholar
[9]	T. Høholdt, J. H. van Lint and R. Pellikaan, On the decoding of algebraic-geometric codes,, IEEE Trans. Inform. Theory, 41 (1995), 1589. doi: 10.1109/18.476214. Google Scholar
[10]	T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes,, in, 1 (1998), 871. Google Scholar
[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. doi: 10.1109/18.108255. Google Scholar
[12]	S. Kampf, Bounds on collaborative decoding of interleaved Hermitian codes with a division algorithm and virtual extension,, 3ICMCTA Special Issue of Designs, (2012). Google Scholar
[13]	S. Kampf, M. Bossert and S. Bezzateev, Some results on list decoding of interleaved Reed-Solomon codes with the extended euclidean elgorithm,, in, (2008), 31. Google Scholar
[14]	S. Kampf, M. Bossert and I. I. Bouw, Solving the key equation for Hermitian codes with a division algorithm,, in, (2011), 1008. Google Scholar
[15]	D. A. Leonard, A generalized Forney formula for algebraic-geometric codes,, IEEE Trans. Inform. Theory, 42 (1996), 1263. doi: 10.1109/18.508855. Google Scholar
[16]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'', North-Holland Mathematical Library, (1988). Google Scholar
[17]	M. O'Sullivan and M. Bras-Amorós, The key equation for one-point codes,, in, 5 (2008), 99. Google Scholar
[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. doi: 10.1109/18.165441. Google Scholar
[19]	R. M. Roth, "Introduction to Coding Theory,'', Cambridge University Press, (2006). Google Scholar
[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. doi: 10.1109/18.476240. Google Scholar
[21]	B.-Z. Shen, Solving a congruence on a graded algebra by a subresultant sequence and its application,, J. Symbolic Comput., 14 (1992), 505. doi: 10.1016/0747-7171(92)90020-5. Google Scholar
[22]	A. Skorobogatov and S. G. Vlăduţ, On the decoding of algebraic-geometric codes,, IEEE Trans. Inform. Theory, IT-36 (1990), 1051. doi: 10.1109/18.57204. Google Scholar
[23]	H. Stichenoth, "Algebraic Function Fields and Codes,'' 2^nd edition,, Springer-Verlag, (2009). Google Scholar
[24]	M. A. Tsfasman and S. G. Vlăduţ, "Algebraic-Geometric Codes,'', Kluwer Academic Publishers Group, (1991). Google Scholar
[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. doi: 10.1002/mana.19821090103. Google Scholar
[26]	K. Yang and P. V. Kumar, On the true minimal distance of Hermitian codes,, in, (1992), 99. doi: 10.1007/BFb0087995. Google Scholar

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References:

[1]	P. Beelen and T. Høholdt, The decoding of algebraic geometry codes,, in, 5 (2008), 99. Google Scholar
[2]	D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry,'', Springer, (1998). Google Scholar
[3]	I. M. Duursma, Algebraic geometry codes: general theory,, in, 5 (2008), 99. Google Scholar
[4]	D. Ehrhard, "Über das Dekodieren algebraisch-geometrischer Codes,'' Dissertation,, Universität Düsseldorf, (1991). Google Scholar
[5]	J. I. Farrán, Decoding algebraic geometry codes by the key equation,, Finite Field Appl., 6 (2000), 207. doi: 10.1006/ffta.1999.0274. Google Scholar
[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. doi: 10.1109/18.179340. Google Scholar
[7]	V. D. Goppa, Codes on algebraic curves,, Dokl. Akad. Nauk SSSR, 259 (1981), 1289. Google Scholar
[8]	V. Guruswami and M. Sudan, Improved decoding of Reed-Solomon and algebraic-geometric codes,, in, (1998). Google Scholar
[9]	T. Høholdt, J. H. van Lint and R. Pellikaan, On the decoding of algebraic-geometric codes,, IEEE Trans. Inform. Theory, 41 (1995), 1589. doi: 10.1109/18.476214. Google Scholar
[10]	T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes,, in, 1 (1998), 871. Google Scholar
[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. doi: 10.1109/18.108255. Google Scholar
[12]	S. Kampf, Bounds on collaborative decoding of interleaved Hermitian codes with a division algorithm and virtual extension,, 3ICMCTA Special Issue of Designs, (2012). Google Scholar
[13]	S. Kampf, M. Bossert and S. Bezzateev, Some results on list decoding of interleaved Reed-Solomon codes with the extended euclidean elgorithm,, in, (2008), 31. Google Scholar
[14]	S. Kampf, M. Bossert and I. I. Bouw, Solving the key equation for Hermitian codes with a division algorithm,, in, (2011), 1008. Google Scholar
[15]	D. A. Leonard, A generalized Forney formula for algebraic-geometric codes,, IEEE Trans. Inform. Theory, 42 (1996), 1263. doi: 10.1109/18.508855. Google Scholar
[16]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'', North-Holland Mathematical Library, (1988). Google Scholar
[17]	M. O'Sullivan and M. Bras-Amorós, The key equation for one-point codes,, in, 5 (2008), 99. Google Scholar
[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. doi: 10.1109/18.165441. Google Scholar
[19]	R. M. Roth, "Introduction to Coding Theory,'', Cambridge University Press, (2006). Google Scholar
[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. doi: 10.1109/18.476240. Google Scholar
[21]	B.-Z. Shen, Solving a congruence on a graded algebra by a subresultant sequence and its application,, J. Symbolic Comput., 14 (1992), 505. doi: 10.1016/0747-7171(92)90020-5. Google Scholar
[22]	A. Skorobogatov and S. G. Vlăduţ, On the decoding of algebraic-geometric codes,, IEEE Trans. Inform. Theory, IT-36 (1990), 1051. doi: 10.1109/18.57204. Google Scholar
[23]	H. Stichenoth, "Algebraic Function Fields and Codes,'' 2^nd edition,, Springer-Verlag, (2009). Google Scholar
[24]	M. A. Tsfasman and S. G. Vlăduţ, "Algebraic-Geometric Codes,'', Kluwer Academic Publishers Group, (1991). Google Scholar
[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. doi: 10.1002/mana.19821090103. Google Scholar
[26]	K. Yang and P. V. Kumar, On the true minimal distance of Hermitian codes,, in, (1992), 99. doi: 10.1007/BFb0087995. Google Scholar

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