Generalized minimum-distance decoding of euclidean-space codes and lattices

IEEE Transactions on Information Theory

Volumn 42, Issue 6 PART 1, 1996, Pages 1992-2026

  David Forney Jr , G ^a   Vardy, Alexander ^b  

^a MOTOROLA INC   (United States)

^b University of Illinois at Urbana Champaign   (United States)

Author keywords

Decoding complexity; Effective error coefficient; Generalized minimum distance decoding; Multilevel codes; Multistage decoding; Sphere packings and lattices

Indexed keywords

ALGORITHMS; BLOCK CODES; CODING ERRORS; COMMUNICATION CHANNELS (INFORMATION THEORY); COMPUTATIONAL COMPLEXITY; ENCODING (SYMBOLS); PROBABILITY; RELIABILITY THEORY; TRELLIS CODES;

DECODING COMPLEXITY; EFFECTIVE ERROR COEFFICIENT; EUCLIDEAN SPACE CODES AND LATTICES; GENERALIZED MINIMUM DISTANCE DECODING; MULTILEVEL CODES; MULTISTAGE DECODING;

DECODING;

EID: 0030286484     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/18.556693     Document Type: Article

References (45)

1. O. Amrani, Y. Be'ery, A. Vardy, F.-W. Sun, and H. C. A. van Tilborg, "The Leech lattice and the Golay code: bounded-distance decoding and multilevel constructions," IEEE Trans. Inform. Theory, vol. 40, pp. 1030-1043, 1994.
2. K. Araki, M. Takada, and M. Morii, "The generalized syndrome polynomial and its application to the efficient decoding of ReedSolomon codes based on the GMD criterion," Proc. IEEE Int. Symp. on Information Theory (San Antonio, TX, 1993), p. 34.
3. E. R. Berlekamp, "Codes and games," Shannon Lecture, Proc. IEEE Int. Symp. on Information Theory (San Antonio, TX, 1993).
4. E. R. Berlekamp, "Bounded distance +1 soft-decision Reed-Solomon decoding," IEEE Trans. Inform. Theory, vol. 42, pp. 704-720, 1996.
5. E. L. Blokh and V. V. Zyablov, "Coding of generalized cascade codes," Probl. Pered. Inform., vol. 10, pp. 45-50, 1974 (in Russian).
6. A. R. Calderbank, "Multilevel codes and multistage decoding," IEEE Trans. Commun., vol. 37, pp. 222-229, 1989.
7. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups. New York: Springer-Verlag, 1988.
8. M. A. O. de Costa e Suva and R. Palazzo, Jr., "A bounded-distance decoding algorithm for lattices obtained from a generalized code formula," - IEEE Trans. Inform. Theory, vol. 40, pp. 2075-2082, 1994.
9. G. Einarsson and C. E. Sundberg, "A note on soft decision decoding with successive erasures," IEEE Trans. Inform. Theory, vol. IT-22, pp. 88-96, 1976.
10. M. Elia, "Algebraic decoding of the (23,12,7) Golay code," IEEE Trans. Inform. Theory, vol. IT-33, pp. 150-151, 1987.
11. N. D. Elkies, personal communication, Mar. 1995.
12. G. D. Forney, Jr., "Generalized minimum distance decoding," IEEE Trans. Inform. Theory, vol. IT-12, pp. 125-131, 1966.
13. "Coset codes I: Introduction and geometrical classification," IEEE Trans. Inform. Theory, vol. 34, pp. 1123-1151, 1988.
14. "Coset codes II: Binary lattices and related codes," IEEE Trans. Inform. Theory, vol. 34, pp. 1152-1187, 1988.
15. "A bounded distance decoding algorithm for the Leech lattice, with generalizations," IEEE Trans. Inform. Theory, vol. 35, pp. 906-909, 1989.
16. "Geometrically uniform codes," IEEE Trans. Inform. Theory, vol. 37, pp. 1241-1260, 1991.
17. G. D. Forney, Jr., M. D. Trott, and S.-Y. Chung, "On the capacity of coset codes and multilevel coset codes," Proc. IEEE Int. Symp. on Information Theory and Applications (Victoria, BC, Canada, 1996).
18. M. Fossorier and S. Lin, "Reliability-based decoding of binary linear codes viewed as bounded-distance decoding," IEEE Trans. Inform. Theory, submitted for publication, 1996.
19. H. Imai and S. Hirakawa, "A new multilevel coding method using errorcorrecting codes," IEEE Trans. Inform. Theory, vol. IT-23, pp. 371-377, 1977.
20. D. E. Knuth, The Art of Computer Programming, vol. 3: Sorting and Searching. Reading, MA: Addison-Wesley, 1973.
21. M. S. Kolluru, G. L. Feng, and T. R. N. Rao, "One run generalized minimum distance decoding of algebraic geometric codes," preprint.
22. Y. Kofman, E. Zehavi, and S. Shamai, "Performance analysis of a multilevel coded modulation system," IEEE Trans. Commun., vol. 42, pp. 299-312, 1994.
23. R. Kotier, "A new efficient error-erasure location scheme in GMD de-coding," Proc. IEEE Int. Symp. on Information Theory (San Antonio, TX., 1993), p. 33.
24. "Fast generalized minimum-distance decoding of algebraicgeometry and Reed-Solomon codes," IEEE Trans. Inform. Theory, vol. 42, pp. 721-737, 1996.
25. J. Leech and N. J. A. Sloane, "Sphere packings and error-correcting codes," Canad. J. Math., vol. 23, pp. 718-745, 1971.
26. S.'Lin and D. J. Costello, Jr., Error Control Coding: Fundamentals and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1983.
27. F. J. Mac Williams and N. J. A. Sloane, The Theory of Error-Correcting Codes. New York: North-Holland, 1977.
28. JJ. Pless, "Decoding the Golay codes," IEEE Trans. Inform. Theory, vol. IT-32, pp. 561-567, 1986.
29. G. J. Pottie and D. P. Taylor, "Multilevel codes based on partitioning," IEEE Trans. Inform. Theory, vol. 35, pp. 87-98, 1989.
30. C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J., vol. 27, pp. 379-423 and pp. 623-656, 1948.
31. B.-Z. Shen, K. K. Tzeng, and C. Wang, "A -bounded-distance decoding algorithm for binary linear block codes achieving the minimum effective error coefficient," this issue, pp. 1987-1991.
32. J. Snyders and Y. Be'ery, "Maximum likelihood soft decoding of binary block codes and decoders for the Golay codes," IEEE Trans. Inform. Theory, vol. 35, pp. 963-975, 1989.
33. U. Sorger, "Fast generalized-minimum-distance decoding," Proc. IEEE Int. Symp. on Information Theory (San Antonio, TX, 1993), p. 30.
34. "A new Reed-Solomon code decoding algorithm based on Newton's interpolation," IEEE Trans. Inform. Theory, vol. 39, pp. 358-365, 1993.
35. F.-W. Sun and H. C. A. van Tilborg, "The Leech lattice, the octacode, and decoding algorithms," IEEE Trans. Inform. Theory, vol. 41, pp. 1097-1106, 1995.
36. D. J. Taipale and M. B. Pursley, "An improvement to generalizedminimum-distance decoding," IEEE Trans. Inform. Theory, vol. 37, pp. 167-172, 1991.
37. D. J. Taipale and M. J. Seo, "An efficient soft-decision ReedSolomon decoding algorithm," IEEE Trans. Inform. Theory, vol. 40, pp. 1130-1139, 1994.
38. V. Tarokh, "Trellis complexity vs. the coding gain of lattice-based communication systems," Ph.D. dissertation, University of Waterloo, Waterloo, Ont., Canada, 1995.
39. A. Vardy, "The Nordstrom-Robinson code: representation over GF(4) and efficient decoding," IEEE Trans. Inform. Theory, vol. 40, pp. 1686-1693, 1994.
40. "Even more efficient bounded-distance decoding of the hexacode, the Golay code, and the Leech lattice," IEEE Trans. Inform. Theory, vol. 41, pp. 1495-1499, 1995.
41. A. Vardy and Y. Be'ery, "Maximum-likelihood decoding of the Leech lattice," IEEE Trans. Inform. Theory, vol. 39, pp. 1435-1444, 1993.
42. U. Wachsmann and J. Huber, "Power and bandwidth efficient digital communication using turbo codes in multilevel codes," European Trans. Telecomm., vol. 6, pp. 557-567, 1995.
43. L. R. Welch and E. R. Berlekamp, "Error correction for algebraic block codes," U.S. Patent 4633470, Dec. 30, 1986.
44. V. A. Zinoviev, "Generalized cascade codes," Probl. Pered. Inform., vol. 12, pp. 5-15, 1976 (in Russian).
45. V. A. Zinoviev, S. N. Litsyn, and S. L. Portnoi, "Concatenated codes in Euclidean space," Probl Pered. Inform., vol. 25, pp. 62-75, 1989 (in Russian).