- 11.4 Error-correction code types g(p)=1+p+ps+p6+p+pp which are both dividers of 1 p23. The corresponding minimum distance is dmin = 7, corresponding to a correction capability of three-bit errors. Maximum-length shift-register codes These are cyclic codes of the form (2m - 1,m) with m 3, which are generated by polynomials of the form g(p) = (1 + pn)/h(p), where h(p) is a primitive polynomial of degree m (meaning an irreducible polynomial dividing 1 1+p9 with =2m 1 being the smallest possible integer). The minimum distance for this code is dmin = 2m-1 , indicating the capability of correcting a maximum of w={2m-1 = - 1/2} simultaneous errors. For instance, the code (15, 4) has a three-bit-error correction capability. The code rate (number of message-payload bits divided by the block length) is R = 4/15 = 26%, which is relatively poor in terms of bandwidth use. The maximum-length code sequences are labeled as "pseudo-noise," owing to their auto-correlation properties, which closely emulate white noise. Bose-Chaudhuri-Hocquenghem (BCH) codes These are cyclic codes of theform(2m-1,k)withm > 3,n-k<mt, - where m is an arbitrary positive integer, and t is the number of errors that the code can correct (not all m, t values being yet eligible). The BCH codes are generated by irreducible polynomials dividing 1 + p2m-1 The first of these polynomials is g(p) = 1 1+p + p2, corresponding to the block (7,4) with t = 1. The minimum distance for BCH codes is dmin=2t+ 3, 5, 7, corresponding to error-correcting capabilities of t= 1, 2, 3, respec- tively. For instance, the block (31, 11) with t = 5 can be corrected for a number of errors corresponding to almost half of the 11 message bits, while the code rate is R = 11/31 = 35.5%. Table 11.4 lists the possible parameter combinations (m, n, k, t) for BCH codes up to m = 6 corresponding to block lengths 7 <n< 63, and their generating polyno- mials, as expressed in octal form.5 Extended lists of BCH codes and polynomials for <8 and m < 34 can be found in.6 5 The octal representation 