of a polynomial is readily understood by the following two examples, showing first the representation of the polynomial coefficients in binary, then in decimal, then, finally, in octal: ((p) = p3 + p + 19decimal Q2(p) = ppp6+p4+1=101101001 = 721decimal 6 J.G. Proakis, Digital Communications (New York: McGraw Hill, 2001), pp. 438-9, and other related references, therein. Error correction Table 11.4 List of BCH codes with block lengths n </ 63, and their generating polynomials g(p), expressed in octal. 12471 1701317 166623567 1033500423 157464165547 17323260404441 107657 1363026512351725 5423325 6331141367235453 313365047 472622305527250155 5231045543503271737 Reed-Solomon (RS) codes These represent a subfamily of BCH codes based on a specific arrangement, noted RS(N, K). The code block is made of N symbols comprising K message symbols and N - K parity symbols. The message or parity symbols of length m are nonbinary (e.g., m = 8 for byte symbols). The total block length is, thus, Nm. The RS code format is RS(N = 2m - 1, K=N-2t). Its minimum distance is dmin = N - K + 1, corresponding to a symbol error-correction capability of t = (N - K)/2. The code rate is R = (N - 2t)/N = 1 - 2t/N. For instance, the RS(255, 231) code having N = 255 symbols (m = 8) and t = 12 (K = 231) corresponds to a code rate of 90% (or a relatively small bandwidth expansion factor of N/K - 1 = 10.4%. Thus, symbol errors can be corrected up to about 1/20 of the block length with only N - K = 24 symbols, representing an ECC overhead of nearly 10%. This example illustrates the power of RS codes and justifies their widespread use in telecommunications. Concatenated block codes It is possible to concatenate, or use two different ECCs successively. This is usually done with a nonbinary ECC (outer code), which is labeled (N, K) and a binary ECC (inner code), which is labeled (n, k). The message coding begins with the outer code and ends with the inner code, yielding a block of the form (n N, kK), meaning that each of the K nonbinary symbols is encode