On the rectangularity of nonlinear block codes

IEEE Transactions on Information Theory

Volumn 45, Issue 2, 1999, Pages 720-725

  Sidorenko, Vladimir ^a   Martin, Lan ^a   Honary, Bahram ^a  

^a INSTITUTE FOR INFORMATION TRANSMISSION PROBLEMS   (Russian Federation)

Author keywords

Codes; Delsarte goethals; Hadamard; Kerdock; Levenshtein; Nonlinear; Nordstrom robinson; Rectangular; Trellis

Indexed keywords

COMPUTATIONAL GEOMETRY; DECISION THEORY; DECODING; MAXIMUM LIKELIHOOD ESTIMATION; TRELLIS CODES;

DELSARTE-GOETHALS CODE; HADAMARD CODE; KERDOCK CODE; LEVENSHTEIN CODE; NORDSTROM-ROBINSON CODE;

BLOCK CODES;

EID: 0033100442     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/18.749021     Document Type: Article

References (27)

1. L.R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rate," IEEE Trans. Inform. Theory, vol. IT-20, pp. 284-287, Mar. 1974.
2. P. Delsarte and J. M. Goethals, "Alternating bilinear forms over GF(q)," J. Combin. Theory, Ser. A, vol. 19, pp. 26-50, 1975.
3. U. Dettmar, R. Raschofer, and U. K. Sorger, "On the trellis complexity of block and convolutional codes," Problems Inform. Transm., vol. 32, no. 2, pp. 10-21, Apr./June 1996.
4. G.D. Forney Jr., "Coset codes-Part II: Binary lattices and related codes," IEEE Trans. Inform. Theory, vol. 34, pp. 1152-1187, Sept. 1988.
5. G.D. Forney Jr. and M. D. Trott, "The dynamics of group codes: State spaces, trellis diagrams, and canonical encoders," IEEE Trans. Inform. Theory, vol. 39, pp. 1491-1513, Sept. 1993.
6. J. Hadamard, "Resolution d'une question relative aux déterminants," Bull. Sei. Math., vol. 17, no. 2, pp. 240-248, 1893.
7. A.R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, "The ,£4-linearity of Kerdock, Preparata, Goethals, and related codes," IEEE Trans. Inform. Theory, vol. 40, pp. 301-319, Mar. 1994.
8. A.M. Kerdock, "A class of low-rate nonlinear binary codes," Inform. Contr., vol. 20, pp. 182-187, 1972.
9. A.D. Kot and C. Leung, "On the construction and dimensionality of linear block code trellises," in Proc. IEEE Int. Symp Inform. Theory, San Antonio, TX, Jan. 1993, p. 291.
10. F.R. Kschischang and V. Sorokine, "On the trellis structure of block codes," IEEE Trans. Inform. Theory, vol. 41, pt. 2, pp. 1924-1937, Nov. 1995.
11. F.R. Kschischang, "The trellis structure of maximal fixed-cost codes," IEEE Trans. Inform. Theory, vol. 42, pt. 1, pp. 1828-1838, Nov. 1996.
12. V.I. Levenshtein, "The application of Hadamard matrices to a problem in coding," Probl. Kibern., vol. 5, pp. 123-136, 1961; Problems of Cybernetics, vol. 5, pp. 166-184, 1964 (Engl. transi.).
13. F.J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977.
14. J.L. Massey, "Foundations and methods of channel coding," in Proc. Int. Conf. Inform. Theory Syst., NTG-Fachberichte, Berlin, Germany, Sept. 1978, vol. 65, pp. 148-157.
15. R.J. McEliece, "On the BCJR trellis for linear block codes," IEEE Trans. Inform. Theory, vol. 42, pp. 1072-1092, July 1996.
16. D.J. Muder, "Minimal trellises for block codes," IEEE Trans. Inform. Theory, vol. 34, pp. 1049-1053, Sept. 1988.
17. A.A. Nechaev, "The Kerdock code in a cyclic form," Diskret. Math., vol. 1, pp. 123-139, 1989; Diskrete Math. Appl., vol. 1, pp. 365-384, 1991 (Engl. transi.).
18. A.W. Nordstrom and J. P. Robinson, "An optimum nonlinear code," Inform. Contr., vol. 11, pp. 613-616, 1967.
19. R.E. A. C. Paley, "On orthogonal matrices," J. Math. Phys., vol. 12, pp. 311-320, 1933.
20. V. Sidorenko, "The Euler characteristic of the minimal code trellis," Problems Inform. Transm., vol. 33, no. 1, pp. 87-93, Jan./Mar. 1997.
21. V. Sidorenko, G. Markarian, and B. Honary, "Minimal trellis design for linear codes based on the Shannon product," IEEE Trans. Inform. Theory, vol. 42, pt. 1, pp. 2048-2053, Nov. 1996.
22. N.J. A. Sloane and J. J. Seidel, "A new family of nonlinear codes obtained from conference matrices," Ann. NY Acad. Sei., Article 1, vol. 175, pp. 363-365, 1970.
23. A. Vardy and F. R. Kschischang, "Proof of a conjecture of McEliece regarding the optimality of the minimal trellis," IEEE Trans. Inform. Theory, vol. 42, pt. 1, pp. 2027-1834, Nov. 1996.
24. A.J. Viterbi, "Error bounds for convolutional codes and an asymptotically optimum decoding algorithm," IEEE Trans. Inform. Theory, vol. 13, pp. 260-269, Apr. 1967.
25. J.C. Willems, "Models for dynamics," in Dynamics Reported, U. Kirchgraber and H. O. Walther, Eds. New York: Wiley, 1989, vol. 2.
26. J.K. Wolf, "Efficient maximum likelihood decoding of linear block codes using a trellis," IEEE Trans. Inform. Theory, vol. IT-24, pp. 76-80, Jan. 1978.
27. V. Zyablov and V. Sidorenko, "Bounds on complexity of trellis decoding of linear block codes," Problems Inform. Transm., vol. 29, no. 3, pp. 3-9, July-Sept. 1993.