B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

866 ERROR CORRECTING CODES
11. }-I. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I, Wiley InterScience, 2001
(reprint), Hoboken, NJ.
12. C. Berrou, A. Glavieux, and P. Thitimajshima, "Near Shannon Limit Error Correcting Coding and
Decoding: Turbo Codes," Proc. 1993 IEEE International Conference on Communications, pp.
1064-1070, Geneva, Switzerland, May 1993.
13. L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, "Optimum Decoding of Linear Codes for Minimizing
Symbol Error Rate," IEEE Trans. Inform. Theory, vol. IT-20, pp. 284-287, March 1974.
14. C. Berrou and A. Glavieux, "Near Optimum Error Correcting Coding and Decoding: Turbo-Codes,"
IEEE Trans. Commun., vol. 44, no. 10, pp. 1261-1271, Oct. 1996.
15. R. G. Gallager, Low Density Parity Check Codes, Monograph, MIT Press, cambridge, MA, 1963.
16. David J. C. MacKay and R. M. Neal, "Good Codes Based on Very Sparse Matrices," Fifth IMA
Conference on Cryptography and Coding, Lecture Notes on Computer Science no. 1025. Colin
Boyd, Ed., Springer, Berlin, 1995 pp. 100-111.
17. D. J. C. MacKay and R. M. Neal, "Near Shannon Limit Performance of Low Density Parity Check
Codes," Electron. Lett. , vol. 33, March 13, 1997.
18. R. M. Tanner, "A Recursive Approach to Low Complexity Codes," IEEE Trans. Inform. Theory,
IT-27, pp. 533-547, Sept. 1981.
PROBLEMS
14.1-1 Golay's (23, 12) codes are three-error correcting codes. Verify that n = 23 and k = 12 satisfies
the Hamming bound exactly for t = 3.
14.1-2 (a) Determine the Hamming bound for a ternary code (whose three code symbols are 0, I, 2).
(b) A ternary (11, 6) code exists that can correct up to two errors. Verify that this code satisfies
the Hamming bound exactly.
14.1-3 Confirm the possibility of a (18, 7) binary code that can correct up to three errors. Can this
code correct up to four errors?
14.2-1 If G and H are the generator and parity check matrices, respectively, then show that
14.2-2 Given a generator matrix
GH 1 =0
G = [I I ]
construct a (3, I) code. How many errors can this code correct? Find the codeword for data
vectors d = 0 and d = 1. Comment.
14.2-3
Repeat Prob. 14.2-2 for
G = (1 l ]
14.2-4
This gives a (5, 1) code.
A generator matrix
G=[ I 0 I 0 0 ]