n Fig. 11.7, where Reed-Solomon codes RS(255, k) are used. As the figure shows, the implementation of a single RS(255, 239) code, which has a redundancy of 255/239 - 1 = 6.7%, provides a coding gain of about 1.5 dB at BER = 10-4. The corrected BER is, however, less than BER = 10-12 When this RS 15 Experimental data from: O. Ait Sab and J. Fang, Concatenated forward error correction schemes for long- haul DWDM optical transmission systems. In Proc. European Conference on Optical Communications, ECOC'99, Vol. II (1999), p. 290. Error correction code is concatenated twice, i.e., using RS(255,239) + RS(255, 239) with a redundancy of (255/239)2-1 1= 13.8%, the coding gain is 5.4 dB at BER = 10-5 Alternatively, one can increase the number of parity bits in the inner code, i.e., using RS(255, 223), which corresponds to the concatenated code RS(255, 239) + RS(255, 223), which has a redundancy of 2552/(239 x 223) - 1= 22.0%. The resulting coding gain is seen to improve with respect to the previous ECC concatenation, corresponding to 5.8 dB at BER 10-5. We also note from the figure that the BER is reduced by more than two orders of magnitude. Such data illustrate the powerful effect of ECC on error control. As I have mentioned earlier in this chapter, the BER is always nonzero, meaning that there is no such thing as an absolute error correction or BER= 0.0 X 100 to infinite accuracy. Two key questions are: (a) how much a corrected BER can be made close to zero, and (b) whether there is a SNR (or limit under which ECCs are of limited effect in any BER correction? Such questions are answered in Chapter 13, through one of the most famous Shannon theorems. Exercises 11.1 (B): Tabulate the block codewords of the Hamming code (n,k) (7, 4) defined by the following generator matrix: 61000101 0100111 0010010 0001110 11.2 (B): Assuming the Hamming code (n,k)= (7, 4) with the parity-check matrix 1011100 = 1101010 1110001 determine the single errors and corrected codewords from the three received blocks that are defined as follows: Z1