Soft Decoding Techniques for Codes and Lattices, Including the Golay Code and the Leech Lattice

IEEE Transactions on Information Theory

Volumn 32, Issue 1, 1986, Pages 41-50

  Conway, John H ^a   Sloane, N J A ^b  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^b LUCENT TECHNOLOGIES   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER PROGRAMMING - ALGORITHMS; INFORMATION THEORY; SIGNAL PROCESSING - DIGITAL TECHNIQUES;

GOLAY CODE; LEECH LATTICE; SOFT DECODING TECHNIQUES;

CODES, SYMBOLIC;

EID: 0022563696     PISSN: 00189448     EISSN: 15579654     Source Type: Journal    
DOI: 10.1109/TIT.1986.1057135     Document Type: Article

References (71)

1. G. Battail, “Décodage pondere optimal des codes lineaires en blocs. I. Emploi simplifie du diagramme du trellis,” Ann. Telecommun., vol. 38, 443–459, 1983.
2. G. Battail, M. C. Decouvelaere, and P. Godlewski, “ Replication decoding,” IEEE Trans. Inform. Theory, vol. IT-25, pp. 332–345, 1979.
3. L. D. Baumert and R. J. McEliece, “Performance of some block codes on a Gaussian channel,” in Proc. 1975 Int. Telemetering Conf, pp. 189–195.
4. ——, “Soft decision decoding of block codes,” Jet Propulsion Lab., Calif. Inst. Tech., Deep Space Network Progress Rep. 42–47, July 1978, pp. 60–64.
5. L. D. Baumert, R. J. McEliece, and G. Solomon, “Decoding with multipliers,” Jet Propulsion Lab., Calif. Inst. Tech., Deep Space Network Progress Rep. 42–34, Aug. 1976, pp. 43–46.
6. T. Berger, “Optimum quantizers and permutation codes,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 759–765, 1972.
7. T. Berger, F. Jelinek, and J. K. Wolf, “Permutation codes for sources,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 160–169, 1972.
8. E. R. Berlekamp, “The technology of error-correcting codes,” Proc. IEEE, vol. 68, pp. 564–593, 1980.
9. R. E. Blahut, Theory and Practice of Error Control Codes. Reading, MA: Addison-Wesley, 1983, especially p. 482.
10. I. F. Blake, “The Leech lattice as a code for the Gaussian channel,” Inform. Contr., vol. 19, pp. 66–74, 1971.
11. R. W. D. Booth, M. A. Herro, and G. Solomon, “Convolutional coding techniques for certain quadratic residue codes,” in Proc. 1975 Int. Telemetering Conf., pp. 168–177.
12. P. de Buda, “Encoding and decoding algorithms for an optimal lattice-based code,” in IEEE Conf. Rec. 81CH1648-5, Int. Conf. Commun., June 1981, pp. 65.3.1-65.3.5.
13. R. de Buda, “The upper error bound of a new near-optimal code,” IEEE Trans. Inform. Theory, vol. IT-21, pp. 441–445, 1975.
14. D. Chase, “A class of algorithms for decoding block codes with channel measurement information,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 170–182, 1972.
15. J. H. Conway, “Three lectures on exceptional groups,” in Finite Simple Groups, M. B. Powell and G. Higmann, Eds. New York: Academic, 1971, pp. 215–247.
16. J. H. Conway, “The Golay codes and the Mathieu groups,” in The Leech Lattice, Sphere Packings, and Related Topics, J. H. Conway and N. J. A. Sloane. New York: Springer-Verlag. (in preparation).
17. J. H. Conway, R. A. Parker, and N. J. A. Sloane, “The covering radius of the Leech lattice,” Proc. Royal Soc. London, Ser. A, vol. 380, pp. 261–290, 1982.
18. J. H. Conway and V. Pless, “On the enumeration of self-dual codes,” J. Combinatorial Theory, vol. 28A, pp. 26–53, 1980.
19. J. H. Conway and N. J. A. Sloane, “On the enumeration of lattices of determinant one,” J. Number Theory, vol. 15, pp. 83–94, 1982.
20. —, “The unimodular lattices of dimension up to 23 and the Minkowski-Siegel mass constants,” Europ. J. Combinatorics, vol. 3, pp. 219–231, 1982.
21. —, “Voronoi regions of lattices, second moments of polytopes, and quantization,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 211–226, 1982.
22. —, “Fast quantizing and decoding algorithms for lattice quantizers and codes,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 227–232, 1982.
23. —, “Twenty-three constructions for the Leech lattice,” Proc. Royal Soc. London, Ser. A, vol. 381, pp. 275–283, 1982.
24. —, “Lorentzian forms for the Leech lattice,” Bull. Amer. Math. Soc., vol. 6, pp. 215–217, 1982.
25. —, “Laminated lattices,” Ann. Math., vol. 116, pp. 593–620, 1982.
26. —, “Complex and integral laminated lattices,” Trans. Amer. Math. Soc., vol. 280, pp. 463–490, 1983.
27. —, “A fast encoding method for lattice codes and quantizers,” IEEE Trans. Inform. Theory, vol. IT-29, pp. 820–824, 1983.
28. —, “On the Voronoi regions of certain lattices,” SIAM J. Algebraic Discrete Methods, vol. 5, pp. 294–305, 1984.
29. ——, The Leech Lattice, Sphere Packings, and Related Topics. New York: Springer-Verlag, in preparation.
30. H. S. M. Coxeter, Regular Polytopes, 3rd ed. New York: Dover, 1973.
31. P. Delsarte, “Partial-optimal piecewise decoding of linear codes,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 70–75, 1978.
32. B. G. Dorsch, “A decoding algorithm for binary block codes and J-ary output channels,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 391–394, 1974.
33. G. S. Evseev, “Complexity of decoding for linear codes”.(in Russian), Probl. Peradach. Inform:, vol. 19, no. 1, pp. 3–8, 1983. English translation in Probl. Inform. Transmission, vol. 19, no. 1, pp. 1–6, 1983.
34. G. D. Forney, Jr., “Generalized minimum distance decoding,” IEEE Trans. Inform. Theory, vol. IT-12, pp. 125–131, 1966.
35. —, “The Viterbi algorithm, “Proc. IEEE, vol. 61, pp. 268–278, 1973.
36. A. Gersho, “Asymptotically optimal block quantization,” IEEE Trans. Inform. Theory, vol. IT-25, pp. 373–380, 1979.
37. A. Gersho, “On the structure of vector quantizers,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 157–166, 1982.
38. E. N. Gilbert, “Gray codes and paths on the n-cube,” Bell Syst. Tech. J., vol. 37, pp. 815–826, 1958.
39. D. M. Gordon, “Minimal permutation sets for decoding the binary Golay code,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 541–543, 1982.
40. R. R. Green, “A serial orthogonal decoder,” JPL Space Programs Summary, vol. 37-39-IV, pp. 247–253, 1966.
41. ——, “Analysis of a serial orthogonal decoder,” JPL Space Programs Summary, vol. 37-53-III, pp. 185–187, 1968.
42. C. M. Hackett, “An efficient algorithm for soft decision decoding of the (24,12) extended Golay code,” IEEE Trans. Commun., vol. COM-29, pp. 909–911, 1981 and vol. COM-30, p. 554, 1982.
43. C. R. P. Hartmann and L. D. Rudolph, “An optimum symbol-by-symbol decoding rule for linear codes,” IEEE Trans. Inform. Theory, vol. IT-22, pp. 514–517, 1976.
44. T. Y. Hwang, “Decoding linear block codes for minimizing word error rate, IEEE Trans. Inform. Theory,” vol. IT-25, pp. 733–737, 1979.
45. —, “Efficient optimal decoding of linear block codes,” IEEE Trans. Inform. Theory, vol. IT-26, pp. 603–606, 1980.
46. D. E. Knuth, The Art of Computer Programming, vol. 3. Reading, MA: Addison-Wesley, 1973, p. 216.
47. J. Leech, “Notes on sphere packings,” Canad. J. Math., vol. 19, pp. 251–267, 1967.
48. J. Leech and N. J. A. Sloane, “Sphere packing and error-correcting codes,” Canad. J. Math., vol. 23, pp. 718–745, 1971.
49. J. I. Lepowsky and A. E. Meurman, “An Eg-approach to the Leech lattice and the Conway group,” J. Algebra, vol. 77, pp. 484–504, 1982.
50. F. J. Mac Williams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977.
51. J. L. Massey, “Foundation and method of channel encoding,” in Proc. Int. Conf. on Info. Theory and Systems, NTG-Fachberichte (Berlin), vol. 65, 1978, pp. 148–157.
52. H. Miyakawa and T. Kaneko, “Decoding algorithm for error-correcting codes by use of analog weights,” Electron. Commun. Japan, vol. 58-A, 18–27, 1975.
53. H. V. Niemeier, “Definite quadratischer Formen der Dimension 24 und Diskriminante 1,” J. Number Theory, vol. 5, pp. 142–178, 1973.
54. J. Paluszkiewicz, R. Stasinski, and Z. Szymanski, “An algorithm for soft-decision decoding of the Golay and other linear codes,” in 1983 IEEE Int. Symp. Information Theory, Abstracts of Papers. New York: IEEE Press, 1983, p. 137.
55. M. Phister, Jr., Logical Design of Digital Computers. New York: Wiley, 1960, pp. 232–234, 399–401.
56. V. Pless, “The children of the (32,16) doubly even codes,” IEEE Trans. Inform. Theory, vol. IT-24, 738–746, 1978.
57. V. Pless and N. J. A. Sloane, “On the classification and enumeration of self-dual codes,” J. Combinatorial Theory, vol. 18, pp. 313–335, 1975.
58. E. C. Posner, “Combinatorial structures in planetary reconnaissance,” in Error Correcting Codes, H. B. Mann, Ed. New York: Wiley, 1969, pp. 15–46.
59. E. M. Reingold, J. Nievergelt, and N. Deo, Combinatorial Algorithms: Theory and Practice. Englewood Cliffs, NJ: Prentice-Hall, 1977.
60. L. D. Rudolph, C. R. P. Hartmann, T. Y. Hwang, and N. Q. Due, “Algebraic analog decoding of lineary binary codes,” IEEE Trans. Inform. Theory, vol. IT-25, pp. 430–440, 1979.
61. S. S. Ryskov and E. P. Baranovskii, “Solution of the problem of least dense lattice coverings of five-dimensional space by equal spheres” (in Russian), Dokl. Akad. Nauk SSSR, vol., 222, pp. 39–42, 1975. English translation in Soviet Math. Doklady, vol. 6, pp. 586–590, 1975.
62. D. Slepian, “Permutation modulation,” Proc. IEEE, vol. 53, pp. 228–236, 1965.
63. N. J. A. Sloane, “Binary codes, lattices and sphere packings,” in Combinatorial Surveys, P. J. Cameron, Ed. New York: Academic, 1977, pp. 117–164.
64. —, “A note on the Leech lattice as a code for the Gaussian channel,” Inform. Contr., vol 46, 270–272, 1980.
65. —, “Tables of sphere packings and spherical codes,” IEEE Trans. Inform. Theory, vol. IT-27, pp. 327–338, 1981.
66. G. Solomon and H. C. A. van Tilborg, “A connection between block and convolutional codes,” SIAM J. Appl. Math., vol. 37, pp. 358–369, 1979.
67. N. N. Tendolkar and C. R. P. Hartmann, “Generalization of Chase algorithms for soft decision decoding of binary linear codes,” IEEE Trans. Inform. Theory, vol. IT-29, 714–721, 1984.
68. A. J. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm,” IEEE Trans. Inform. Theory, vol. IT-13, pp. 260–269, 1967.
69. J. K. Wolf, “Efficient maximum likelihood decoding of linear block codes using a trellis,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 76–80, 1978.
70. J. Wolfmann, “Nouvelles methodes de decodage du code de Golay (24, 12, 8),” Rev. CETHEDEC Cahier, no. 2, pp. 79–88, 1981.
71. J. Wolfmann, “A permutation decoding of the (24, 12, 8) Golay code,” IEEE Trans. Inform. Theory, vol. IT-29, pp. 748–750, 1983.