rror correcting codes. In this case, the syndrome correspond- decoding. This is of value if the receiver can under some circum- ing to an error in the first (most significant) bit of the codeword is stances signal the loss of a received digit. If such a code has distance calculated. The error pattern detector is then simply a combinational d, it can correct combinations of 1 errors and erasures provided that logic circuit which outputs a 1 when it detects this pattern in the shift register. For multiple error correcting decoders, as discussed in the While RS codes are naturally suited to non-binary digits they can following section, the design of the error pattern detector is more sometimes be used very effectively for binary channels by treating complex. groups of m bits as 2m -ary digits. It has been shown that such codes outperform binary codes with the same rate and block length at low 14.5.3 BCH codes output error rates. Also, when used in this way RS codes have a natural burst-error correcting ability, because, for example, a burst BCH codes are the most extensive and powerful family of error-cor- of up to m bit errors will affect at most 2 'digits'. recting cyclic codes, and because of their mathematical structure allow decoders to be implemented reasonably easily, even for 14.5.5 Other cyclic codes multiple error correction. Both binary and non-binary BCH codes exist; there are two classes of binary BCH codes: primitive BCH 14.5.5.1 Majority logic decodable codes codes, which have a block length n = 2m and non-primitive BCH codes, where n is a factor of 2m 1. For any positive integers m and These codes are slightly inferior to BCH codes in terms of error t(t<<2m-1) there exists a primitive BCH code with the parameters correction, but have simple decoder implementations in which the n and r given by Equations 14.24 and 14.25 and minimum distance error pattern detector circuit is a combination of XOR and majority by Equation 14.26. logic gates. (14.24) 14.5.5.2 Burst error correcting codes r=n-ksmt (14.25) 