Improved decoding of reed-solomon and algebraic-geometry codes

IEEE Transactions on Information Theory

Volumn 45, Issue 6, 1999, Pages 1757-1767

  Guruswami, Venkatesan ^a   Sudan, Madhu ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Algebraic geometry codes; Decoding algorithms; Error correcting codes; List decoding; Polynomial time algorithms; Reed solomon codes

Indexed keywords

ALGORITHMS; ASYMPTOTIC STABILITY; CURVE FITTING; ERROR CORRECTION; INFORMATION THEORY; POLYNOMIALS;

ALGEBRAIC GEOMETRY CODES; DECODING ALGORITHMS; ERROR CORRECTING CODES; LIST DECODING; POLYNOMIAL TIME ALGORITHMS; REED-SOLOMON CODES;

DECODING;

EID: 0033183669     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/18.782097     Document Type: Article

References (32)

1. S. Ar, R. Lipton, R. Rubinfeld, and M. Sudan, "Reconstructing algebraic functions from mixed data," SIAM J. Comput, vol. 28, no. 2, pp. 488-511, 1999.
2. E. R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, 1968.
3. "Bounded distance +1 soft-decision Reed-Solomon decoding," IEEE Trans. Inform. Theory, vol. 42, pp. 704-720, May 1996.
4. R. E. Blahut, Theory and Practice of Error Control Codes. Reading, MA: Addison-Wesley, 1983.
5. H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138. Berlin, Germany: Springer Verlag, 1993.
6. I. I. Dumer, "Two algorithms for the decoding of linear codes," Probl. Inform. Transm., vol. 25, no. 1, pp. 24-32, 1989.
7. P. Elias, "List decoding for noisy channels," Tech. Rep. 335, Res. Lab. Electron., MIT, Cambridge, MA, 1957.
8. "Error-correcting codes for list decoding," IEEE Trans. Inform. Theory, vol. 37, pp. 5-12, 1991.
9. G. L. Feng and K. K. Tzeng, "A generalization of the Berlekamp-Massey algorithm for multisequence shift register synthesis with application to decoding cyclic codes," IEEE Trans. Inform. Theory, vol. 37, pp. 1274-1287, 1991.
10. G. D. Forney, "Generalized minimum distance decoding," IEEE Trans. Inform. Theory, vol. IT-12, pp. 125-131, 1966.
11. S. Gao and M. A. Shokrollahi, "Computing roots of polynomials over function fields of curves," unpublished manuscript, Aug. 1998.
12. P. Gemmell and M. Sudan, "Highly resilient correctors for multivariate polynomials," Inform. Processing Lett., vol. 43, no. 4, pp. 169-174, 1992.
13. O. Goldreich, R. Rubinfeld, and M. Sudan, "Learning polynomials with queries: The highly noisy case," in Proc. 36th Annu. IEEE Symp. Foundations of Computer Science, 1995, pp. 294-303.
14. D. Grigoriev, "Factorization of polynomials over a finite field and the solution of systems of algebraic equations," translated from Zapiski Nauchnykh Seminarov Lenningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova (Akad. Nauk SSSR), vol. 137, pp. 20-79, 1984.
15. T. Høholdt, J. H. van Lint, and R. Pellikaan, "Algebraic geometry codes," in Handbook of Coding Theory, V. S. Pless, W. C. Huffamn, and R. A. Brualdi, Eds. Amsterdam, The Netherlands: Elsevier.
16. J. Justesen, "Bounds on list decoding of MDS codes," unpublished manuscript, Apr. 1998.
17. E. Kaltofen, "A polynomial-time reduction from bivariate to univariate integral polynomial factorization," in Proc. 14th Annu. ACM Symp. Theory of Computing (San Francisco, CA, May 5-7, 1982), pp. 261-266.
18. "Polynomial factorization 1987-1991," in LATIN'92 (Lecture Notes in Computer Science, vol. 583), I. Simon, Ed. Berlin, Germany: Springer-Verlag, 1992, pp. 294-313.
19. F. J. Mac Williams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1981.
20. V. Olshevsky and M. A. Shokrollahi, "A displacement structure approach to efficient list decoding of algebraic geometric codes," in Proc. 31st Annu. ACM Symp. Theory of Computing, 1999, to be published.
21. W. W. Peterson, "Encoding and error-correction procedures for Bose-Chaudhuri codes," IRE Trans. Inform. Theory, vol. IT-6, pp. 459-470, 1960.
22. J. Radhakrishnan, personal communication, Jan. 1996.
23. R. M. Roth and G. Ruckenstein, "Efficient decoding of Reed-Solomon codes beyond half the minimum distance," submitted to IEEE Trans. Inform. Theory, Aug. 1998.
24. M. A. Shokrollahi and H. Wasserman, "Decoding algebraic-geometric codes beyond the error-correction bound," in Proc. 29th Annu. ACM Symp. Theory of Computing, 1998, pp. 241-248.
25. V. M. Sidelnikov, "Decoding Reed-Solomon codes beyond (d -1)/2 and zeros of multivariate polynomials," Probl. Inform. Transm., vol. 30, no. 1, pp. 44-59, 1994.
26. H. Stichtenoth, Algebraic Function Fields and Codes. Berlin, Germany: Springer-Verlag, 1993.
27. M. Sudan, "Decoding of Reed-Solomon codes beyond the errorcorrection bound," J. Complexity, vol. 13, no. 1, pp. 180-193, 1997.
28. "Decoding of Reed-Solomon codes beyond the error-correction diameter," in Proc. 35th Annu. Allerton Conf. Communication, Control and Computing, 1997.
29. M. A. Tsfasman, S. G. Vlädu;, and T. Zink, "Modular curves, Shimura curves, and codes better than the Varshamov-Gilbert bound," Math. Nachrichten, vol. 109, pp. 21-28, 1982.
30. J. H. van Lint, Introduction to Coding Theory. New York: Springer-Verlag, 1982.
31. A. Vardy, "Algorithmic complexity in coding theory and the minimum distance problem," STOC, 1997.
32. L. Welch and E. R. Berlekamp, "Error correction of algebraic block codes," U.S. Patent 4633470, Dec. 1986.