between codewords is d=r++1 (4C.1) and the code is able to correct any pattern of t symbol errors and S symbol erasures for which 2t+s<d. (4C.2) Thus, this code can correct up to to = r/2 errors when there are no erasures and up to so = r erasures when there are no errors. Suppose we have a symmetric Q-ary memoryless channel with conditional probabilities (4C.3) y x x Appendix 4C. Reed-Solomon Code Performance where Ps is the uncoded symbol error probability. The decoded bit error probability is given approximately by (4C.4) where to is the number of symbol errors that can be corrected. Table 4C.1 shows values of the uncoded symbol error probabilities needed for each RS code of block length N = Q = 2m and parameter to = r/2 to achieve decoded bit error rates of 10-3, 10-5, 10-7, 10-9, and 10-11 (The last row for each N, however, does not apply here.) Consider, for example, the RS code whose symbols are eight-bit bytes (m = 8), whose block length is N = 256 symbols and whose redundancy is r = 32. This code can correct all symbol error pat- terns of up to to = 16 errors. Suppose this code must attain a decoded bit error rate of Pb = 10-5, then from Table 4C.1 the maximum uncoded sym- bol error probability is Ps = 0.025. Next consider a memoryless erasure channel with conditional probabili- (4C.5) y = erasure where PE is the probability a symbol is erased. The decoded bit error prob- ability is (4C.6) where SO is the number of erasures that can be corrected. Table 4C.1 also shows values of the erasure probabilities needed for each RS code of block- length N = Q = 2m and parameter So = r to achieve decoded bit error rates of 10-3, 10-5, 10-7, 10-9, and 10-11 Again, for the example with N = 256 and r = 32, up to so = 32 erasures can be corrected. For a required decoded bit error rate of Pb = 10-5, from Table 4C.1 the maximum erasure probability is PE = 0.0651. Table 4C.1 is provided courtesy of Dr. Elwyn Berlekamp [16]. Table 4C.1. Required Probabilities [16] Pb = 10-3 Ph=10-11 to,so 4.60E-2 9.51E-3 2.03E-3 4.37E-4 9.