The Nordstrom-Robinson Code: Representation over GF(4) and Efficient Decoding

IEEE Transactions on Information Theory

Volumn 40, Issue 5, 1994, Pages 1686-1693

  Vardy, Alexander ^a  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

efficient decoding; Nordstrom Robinson code; quadracode

Indexed keywords

ALGORITHMS; CALCULATIONS; COMPUTATIONAL COMPLEXITY; DECODING; INVARIANCE; MATRIX ALGEBRA; VECTOR QUANTIZATION;

BINARY IMAGES; ISOMORPHIC LINEAR CODES; MAXIMUM LIKELIHOOD SOFT DECISION DECODING; NORDSTROM-ROBINSON CODE; QUADRACODE; REED MULLER CODE; WEIGHT DISTRIBUTION;

BINARY CODES;

EID: 0028514913     PISSN: 00189448     EISSN: 15579654     Source Type: Journal    
DOI: 10.1109/18.333895     Document Type: Article

References (33)

1. A. D. Abbaszadeh and C. K. Rushforth, “Efficient maximum likelihood decoding of the extended Nordstrom-Robinson code,' in Proc. 25th Allerton Conf. Commun., Control and Computing (Monticello, IL), Oct. 1987, pp. 598-599.
2. J.-P. Adoul, “Fast CELP coding based on algebraic codes,” in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing (Dallas, TX), April 1987.
3. J.-P. Adoul, “Fast ML decoding algorithm for the Nordstrom-Robinson code,” IEEE Trans. Inform. Theory, vol. 33, 931-933, 1987.
4. O. Amrani, Y. Be'ery, A. Vardy, F.-W. Sun, and H. C. A. van Tilborg, “The Leech lattice and the Golary code: Bounded-distance decoding and multilevel constructions,” IEEE Trans. Inform. Theory, vol. 40, pp. 1030-1043, July 1994.
5. S. Aran, “Efficient soft decision decoding of Reed-Muller codes and related non-linear generalizations,” M.Sc. thesis, Tel-Aviv University, 1989.
6. A. E. Ashikmin and S. Litsyn, “Fast decoding algorithms for first-order Reed-Muller and related codes,” preprint.
7. E. F. Assmus, Jr. and H. F. Mattson, Jr., “Algebraic theory of codes,” Report AFCRL-F19628-69C0068, Air Force Cambridge Research Labs., Bedford, MA., 1969.
8. Y. Be'ery and B. Shahar, “VLSI architectures for soft decoding of the Leech lattice and the Golay codes,” in Proc. IEEE Int. Workshop on Microelectronics in Commun. (Interlaken, Switzerland), March 1991.
9. Y. Be'ery and J. Snyders, “Optimal soft decision block decoders based on Fast Hadamard Transform,” IEEE Trans. Inform. Theory, vol. 32, pp. 355-364, 1986.
10. E. R. Berlekamp, “Coding theory and the Mathieu groups,” Inf. Contr., vol. 18, pp. 40-64, 1971.
11. A. E. Brouwer and T. Verhoeff, “An updated table of minimum-distance bounds for binary linear codes,” IEEE Trans. Inform. Theory, vol. 39, pp. 662-677, 1993.
12. G. D. Cohen, S. Litsyn, A. Vardy, and G. Zemor, “Tilings of binary J. Combin.
13. J. H. Conway and N. J. A. Sloane, “Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice,” IEEE Trans. Inform. Theory, vol. 32, pp. 41-50, 1986.
14. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, New NewYork: Springer-Verlag, 1988.
15. J. H. Conway and N. J. A. Sloane, “Orbit and coset analysis of the Golay and related codes,” IEEE Trans. Inform. Theory, vol. 36, pp. 1038-1050, 1990.
16. M. Elia, C. Losana, and F. Neri, “Note on the complete decoding of Kerdock codes,” IEE Proc., vol. 139, pp. 24-28, 1992.
17. G. D. Forney, Jr., “Coset codes II: Binary lattices and related codes,” IEEE Trans. Inform. Theory, vol. 34, pp. 1152-1187, 1988.
18. G. D. Forney, Jr., N. J. A. Sloane, and M. D. Trott, “The Nordstrom-Robinson code is the binary image of the octacode,” in Proc. DIMACS/IEEE Workshop on Coding and Quantization, October 19-26.
19. J.-M. Goethals, “On the Golay perfect binary code,” J. Combin. Theory, vol. 11, pp. 178-186, 1971.
20. A. R. Hammons and P. V. Kumar, “On the apparent duality of Kerdock and Preparata codes,” in Proc. IEEE Int. Symp. Inform. Theory (San Antonio, TX), Jan. 1993, p. 196.
21. A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, “The Z4linearity of Kerdock, Preparata, Goethals and related codes,” IEEE Trans. Inform. Theory, vol. 40, pp. 301-319, Mar. 1994.
22. T. W. Hungerford, Algebra. New York: Holt, Rinehart and Winston, 1974.
23. S. Litsyn, “Fast algorithms for decoding orthogonal and related codes,” Springer-Verlag Lect. Notes Comput. Science, vol. 539, pp. 39-48, 1991.
24. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. New York: North-Holland, 1977.
25. A. W. Nordstrom and J. P. Robinson, “An optimum nonlinear vol. 11, 613-616, 1968.
26. V. Pless, “Decoding the Golay codes,” IEEE Trans. Inform. Theory, vol. 32, pp. 561-567, 1986.
27. J. P. Robinson, “Analysis of Nordstrom's optimum quadratic code,” in Proc. Hawaii Int. Conf. Syst. Sci. (Honolulu, HI), Sept. 1968, pp. 157-161.
28. N. V. Semakov and V. A. Zinoviev, “Complete and quasi-complete balanced Prob. Inf. Trans., vol. 5, 11-13, 1969.
29. N. J. A. Sloane, S. M. Reddy, and C. L. Chen, “New binary codes,” IEEE Trans. Inform. Theory, vol. 18, pp. 503-510, 1972.
30. S. Snover, “The uniqueness of the Nordstrom-Robinson and the Golay binary codes,” Ph.D. thesis, Michigan State Univ., 1973.
31. F. W. Sun and H. C. A. van Tilborg, “Fast bounded-distance decoding of the Nordstrom-Robinson Code,” pp. 391-398 in Communications and Cryptography: Two Sides of One Tapestry, R. E. Blahut et al., Eds. Boston: Kluwer Academic, 1994.
32. A. Vardy and Y. Be'ery, “More efficient soft-decision decoding of the Golary codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 667-672, 1991.
33. A. Vardy and Y. Be'ery, “Maximum-likelihood decoding of the Leech lattice,” IEEE Trans. Inform. Theory, vol. 39, pp. 1435-1444, 1993.