Algebraic-Geometric Codes and Multidimensional Cyclic Codes: A Unified Theory and Algorithms for Decoding Using Gröbner Bases

IEEE Transactions on Information Theory

Volumn 41, Issue 6, 1995, Pages 1733-1751

  Saints, Keith ^a   Heegard, Chris ^b  

^a QUALCOMM INC   (United States)

^b School of Electrical Engineering Cornell University ^*  (United States)

Author keywords

Algebraic geometric codes; Gr bner bases; multidimensional cyclic codes; Sakata's algorithm; transform methods

Indexed keywords

EID: 84927088892     PISSN: 00189448     EISSN: 15579654     Source Type: Journal    
DOI: 10.1109/18.476246     Document Type: Article

References (32)

1. B. Buchberger, “Gröbner bases: An agorithmic method in polynomial ideal theory,” in Multidimensional Systems Theory: Progress, Directions and Open Problems in Multidimensional Systems, N. K. Bose, Ed. Dordrecht, The Netherlands: Reidel, 1985.
2. D. Cox. J. Little, and D. O'Shea, Ideals, Varieties, and Algorithms. New York: Springer-Verlag, 1992.
3. T. Becker and V. Weispfenning, Gröbner Bases. New York: Springer-Verlag, 1992.
4. W. Fulton, Algebraic Curves: An Introduction to Algebraic Geometry. Redwood City, CA: Addson-Wesley, 1989.
5. C. J. Moreno, Algebraic Curves over Finite Fields. Cambridge, England: Univ. Press, 1991.
6. I. R. Shafarevich, Basic Algebraic Geometry. Berlin, Germany: Springer, 1977.
7. J. Harris, Algebraic Geometry: A First Course. New York: Springer-Verlag, 1992.
8. R. Hartshorne, Algebraic Geometry. New York: Springer, 1977.
9. S. C. Porter, “Decoding codes arising from Goppa's construction on algebraic curves,” Ph.D. dissertation, Yale Univ., New Haven, CT, Dec. 1988.
10. S. C. Porter, B.-Z. Shen, and R. Pellikaan, “Decoding geometric Goppa codes using an extra place,” IEEE Trans. Inform. Theory, vol. 38, pp. 1663–1676, Nov. 1992.
11. V. D. Goppa, “Codes associated with divisors,” Probl. Inform. Transm., vol. 13, pp. 33–39, 1977.
12. —, Geometry and Codes. Dordrecht, The Netherlands: Kluwer, 1988.
13. J. H. van Lint, “Algebraic geometric codes,” in Coding Theory and Design Theory, D. Ray-Chaudhuri, Ed. New York: Springer-Verlag, 1990.
14. J. H. van Lint and G. van der Geer, Introduction to Coding Theory and Algebraic Geometry (DMV Lecture Notes). Basel, Switzerland: Birkhauser, 1988.
15. M. A. Tsfasman and S. G. Vläduţ, Algebraic Geometric Codes. Dordrecht, The Netherlands: Kluwer, 1991.
16. H. Stichtenoth, Algebraic Function Fields and Codes. Berlin, Germany: Springer, 1993.
17. H. Imai, “A theory of two-dimensional cyclic codes,” Inform. Contr., vol. 34, pp. 1–21, 1977.
18. A. Poli and L. Huguet, Error Correcting Codes: Theory and Applications. Hemel Flempstead, England: Prentice-Hall, 1992.
19. K. Saints and C. Heegard, “On hyperbolic cascaded Reed-Solomon codes,” in Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes: Proc. AAECC-10, G. Cohen, T. Mora, and O. Moreno, Eds., no. 673 in Lecture Notes in Computer Science. Berlin, Germany: Springer, 1993.
20. K. Saints and C. Heegard, “Hyperbolic cascaded Reed-Solomon codes,” submitted to IEEE Trans. Inform. Theory, 1995.
21. K. Saints, “Algebraic methods for the encoding and decoding problems for multidimensional cyclic codes and algebraic-geometric codes,” Ph.D. dissertation, Cornell Univ., Ithaca, NY, 1995.
22. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977.
23. S. Miura and N. Kamiya, “Geometric-Goppa codes on some maximal curves and their minimum distance,” in Proc. 1993 IEEE Information Theory Workshop, 1993.
24. D. Polemi, C. J. Moreno, and O. J. Moreno, “Search and construction of good a.g. Goppa codes,” manuscript, 1993.
25. R. E. Blahut, Theory and Practice of Error Control Codes. Reading, MA: Addison-Wesley, 1983.
26. E. R. Berlekamp, Algebraic Coding Theory. New York: McGraw-Hill, 1968.
27. J. L. Massey, “Shift-register synthesis and BCH decoding,” IEEE Trans. Inform. Theory, vol. IT-15, pp. 122–127, Jan. 1969.
28. S. Sakata, “Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array,” J. Symb. Comput., vol. 5, pp. 321–337, 1988.
29. S. Sakata, “Extension of the Berlekamp-Massey algorithm to n dimensions,” Inform. Comput., vol. 84, pp. 207–239, 1989.
30. G.-L. Feng and T. R. N. Rao, “Decoding algebraic-geometric codes up to the designed minimum distance,” IEEE Trans. Inform. Theory, vol. 39, pp. 37–45, Jan. 1993.
31. S. Sakata, “Decoding binary 2-D cyclic codes by the 2-D Berlekamp-Massey algorithm,” IEEE Trans. Inform. Theory, vol. 37, pp. 1200–1203, July 1991.
32. S. Sakata, J. Justesen, Y. Madelung, H. E. Jensen, and T. Høholdt, “Fast decoding of AG-codes up to the designed minimum distance,” Tech. Rep. MAT-1993-12, Tech. Univ. of Denmark, Lyngby, 1993.