Codes on finite geometries

IEEE Transactions on Information Theory

Volumn 51, Issue 2, 2005, Pages 572-596

  Tang, Heng ^a   Xu, Jun ^b   Lin, Shu ^c   Abdel Ghaffar, Khaled A S ^c  

^a PMC Sierra Inc   (United States)

^b MARVELL SEMICONDUCTOR INC   (United States)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Cyclic codes; Euclidean geometry; Low density parity check (LDPC) codes; Majority logic decoding; Projective geometry; Quasi cyclic codes; Sum product algorithm

Indexed keywords

ALGORITHMS; BIT ERROR RATE; COMPUTER SIMULATION; DECODING; ENCODING (SYMBOLS); ITERATIVE METHODS; WHITE NOISE;

CYCLIC CODES; EUCLIDEAN GEOMETRY; LOW DENSITY PARITY CHECK (LDPC) CODES; MAJORITY LOGIC DECODING; PROJECTIVE GEOMETRY; QUASI CYCLIC CODES; SUM PRODUCT ALGORITHM;

CODES (SYMBOLS);

EID: 13444301410     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2004.840867     Document Type: Article

References (50)

1. R. G. Gallager, "Low density parity check codes," IRE Trans. Inf. Theory, vol. IT-8, no. 1, pp. 21-28, Jan. 1962.
2. D. J. C. MacKay and R. M. Neal, "Near Shannon limit performance of low density parity check codes," Electron. Lett., vol. 32, pp. 1645-1646, Aug. 1996.
3. M. Sipser and D. Spielman, "Expander codes," IEEE Trans. Inf. Theory, vol. 42, no. 6, pp. 1710-1722, Nov. 1996.
4. D. Spielman, "Linear-time encodable error-correcting codes," IEEE Trans. Inf. Theory, vol. 42, no. 6, pp. 1723-1731, Nov. 1996.
5. M. C. Davey and D. J. C. MacKay, "Low density parity check codes over GF(q)," IEEE Commun. Lett., vol. 2, no. 6, pp. 165-167, Jun. 1998.
6. D. J. C. MacKay, "Good error-correcting codes based on very sparse matrices," IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 399-432, Mar. 1999.
7. T. Richardson and R. Urbanke, "The capacity of low-density parity-check codes under message-passing decoding," IEEE Trans. Info. Theory, vol. 47, no. 2, pp. 599-618, Feb. 2001.
8. T. Richardson, A. Shokrollahi, and R. Urbanke, "Design of capacity-approaching low density parity check codes," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 619-637, Feb. 2001.
9. S. Y. Chung, G. D. Forney, T. Richardson, and R. Urbanke, "On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit," IEEE Commun. Lett., vol. 5, no. 2, pp. 58-60, Feb. 2001.
10. R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inf. Theory, vol. IT-27, no. 5, pp. 533-547, Sep. 1981.
11. L. D. Rudolph, "Geometric Configuration and Majority Logic Decoding Codes," M.E.E. thesis, Univ. Oklahoma, Norman, OK, 1964.
12. L. D. Rudolph, "A class of majority-logic decodable codes," IEEE Trans. Inf. Theory, vol. IT-13, no. 2, pp. 305-307, Apr. 1967.
13. E. J. Weldon Jr., "Difference-set cyclic codes," Bell Syst. Tech. J., vol. 45, pp. 1045-1055, Sep. 1966.
14. K. J. C. Smith, "Majority decodable codes derived from finite geometries," University of North Carolina, Chapel Hill, NC, vol. 561, Institute of Statistics Mimeo Series, 1967.
15. E. J. Weldon Jr., "Euclidean geometry cyclic codes," in Proc. Combinatorial Mathematics and its Application, R. C. Bose and T. A. Dowling, Eds. Chapel Hill, NC: Univ. North Carolina Press, 1967, pp. 377-388.
16. E. J. Weldon Jr., "New generalizations of the Reed-Muller codes, Part II: Non-primitive codes," IEEE Trans. Inf. Theory, vol. IT-14, no. 2, pp. 199-205, Mar. 1968.
17. J. M. Geothals and P. Delsarte, "On a class of majority-logic decodable codes," IEEE Trans. Inf. Theory, vol. IT-14, no. 2, pp. 182-189, Mar. 1968.
18. E. J. Weldon Jr., "Some results on majority-logic decoding," in Error-Correcting Codes, H. Mann, Ed. New York: Wiley, 1968.
19. T. Kasami, S. Lin, and W. W. Peterson, "Polynomial codes," IEEE Trans. Inf. Theory, vol. IT-14, no. 6, pp. 807-814, Nov. 1968.
20. P. Delsarte, "A geometric approach to a class of cyclic codes," J. Comb. Theory, vol. 6, pp. 340-358, 1969.
21. P. Delsarte, J. M. Goethals, and J. MacWilliams, "On GRM and related codes," Inf. Contr., vol. 16, pp. 403-442, 1970.
22. C. L. Chen, "On majority-logic decoding of finite geometry codes," IEEE Trans. Inf. Theory, vol. IT-17, no. 3, pp. 332-336, May 1971
23. T. Kasami and S. Lin, "On majority-logic decoding for duals of primitive polynomial codes," IEEE Trans. Inf. Theory, vol. IT-17, no. 3, pp. 322-331, May 1971.
24. S. Lin, "Multifold Euclidean geometry codes," IEEE Trans. Inf. Theory, vol. IT-19, no. 4, pp. 537-548, Jul. 1973.
25. S. Lin, "On the number of information symbols in polynomial codes," IEEE Trans. Inf. Theory, vol. IT-18, no. 6, pp. 785-794, Nov. 1972.
26. N. Hamada, "On the p-rank of the incidence matrix of a balanced or partial balanced incomplete block design and its applications to error correcting codes," Hiroshima Math. J., vol. 3, pp. 153-226, 1973.
27. C. R. P. Hartmann, J. B. Ducey, and L. D. Rudolph, "On the structure of generalized finite geometry codes," IEEE Trans. Inf. Theory, vol. IT-20, no. 2, pp. 240-252, Mar. 1974.
28. S. Lin and K. P. Yiu, "An improvement to multifold Euclidean geometry codes," Inf. Contr., vol. 28, no. 3, pp. 221-265, 1975.
29. A. Hocquenghem, "Codes correcteurs d'erreurs," Chiffres, vol. 2, pp. 147-156, 1959.
30. R. C. Bose and D. K. Ray-Chaudhuri, "On a class of error correcting binary group codes," Inf. Contr., vol. 3, pp. 68-79, Mar. 1960.
31. I. S. Reed and G. Solomon, "Polynomial codes over certain finite fields," J. Soc. Indust. Appl. Math., vol. 8, pp. 300-304, Jun. 1960.
32. Y. Kou, S. Lin, and M. Fossorier, "Construction of low density parity check codes: A geometric approach," in Proc. 2nd Int. Symp. Turbo Codes and Related Topics, Brest, France, Sep. 4-7, 2000, pp. 137-140.
33. Y. Kou, S. Lin, and M. Fossorier, "Low-density parity-check codes based on finite geometries: A rediscovery and new results," IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2711-2736, Nov. 2001.
34. R. J. McEliece, D. J. C. MacKay, and J. -F. Cheng, "Turbo decoding as an instance of Pearl's belief propagation algorithm," IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 140-152, Feb. 1998.
35. F. R. Kschisehang, B. J. Frey, and H.-A. Loeliger, "Factor graphs and the sum-product algorithm," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498-519, Feb. 2001.
36. Z. W. Li, L. Chen, and S. Lin, "Efficient encoding of quasi-cyclic LDPC codes," IEEE Trans. Commun., submitted for publication.
37. H. B. Mann, Analysis and Design of Experiments. New York: Dover, 1949.
38. R. D. Carmichael, Introduction to the Theory of Groups of Finite Order. New York: Dover, 1956.
39. A. P. Street and D. J. Street, Combinatorics of Experimental Designs Oxford, U.K.: Oxford Sci., Clarendon, 1987.
40. I. Blake and R. C. Mullin, The Mathematical Theory of Coding. New York: Academic, 1975.
41. S. Lin and D. J. Costello Jr., Error Control Coding: Fundamentals and Applications, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2004.
42. D. E. Muller, "Applications of Boolean algebra to switching circuit design and to error detection," IRE Trans. Electron. Comput., vol. EC-2, pp. 6-12, Sep. 1954.
43. I. S. Reed, "A class of multiple-error-correcting codes and the decoding scheme," IRE Trans. Inf. Theory, vol. IT-4, no. 4, pp. 38-49, Sep. 1954.
44. W. W. Peterson and E. J. Weldon Jr., Error Correcting Codes, 2nd ed. Cambridge, MA: MIT Press, 1972.
45. J. L. Massey, Threshold Decoding, Cambridge, MA: MIT Press, 1963.
46. T. Richardson, "Error floors of LDPC codes," in Proc. Allerton Conf. Communication, Control and Computing, Monticello, IL, Oct. 2003, pp. 1426-1435.
47. P. O. Vontobel and R. Koetter, "Lower bounds on the minimum pseudoweight of linear codes," in Proc. IEEE Int. Symp. Information Theory, Chicago, IL, Jun./Jul. 2004, p. 70.
48. R. M. Tanner, "Minimum-distance bounds by graph analysis," IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 808-821, Feb. 2001
49. H. Tang, J. Xu, Y. Kou, S. Lin, and K. Abdel-Ghaffar, "On algebraic construction of Gallager and circulant low-density parity-check codes," IEEE Trans. Inf. Theory, vol. 50, no. 6, pp. 1269-1279, Jun. 2004.
50. R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, revised ed. New York: Cambridge Univ. Press, 1994.