lt of transmitting Y(p) through a noisy channel. We divide the result by g(p) and put it in the form: Z(p)=q(p)g(p)+s(p), (11.24) An irreducible polynomial, like a prime number, can be divided only by itself or by unity. 11.4 Error-correction code types Table 11.3 Message polynomial coefficients and correspond- ing cyclic codeword polynomial coefficients in the (7, 4) block code example, as corresponding to the generator polynomial g(p) = p3 + -p + 1. Message word Cyclic code X(p) X3X2X1X0 Y(p) y6ysy4y3y2y1yo 0000000 0001011 0010110 0011101 0101100 0100111 0111010 0110001 1011000 1010011 1001110 1000101 1110100 1111011 1100010 1101001 where q(p) and s(p) are the quotient and the remainder of the division, respectively. If there were no errors, we would have Z(p) = Y(p) = X(p)g(p), meaning that the quotient would be the original message, q(p) = X(p), and the remainder would be zero, s(p) = 0. As done previously, we call s(p) the syndrome polynomial. A nonzero syndrome means that errors are present in the received code. As previously shown for the vectors E, S, it is possible to map the error polynomial e(p) into s(p), which makes it possible to associate syndrome and error patterns. Looking at Table 11.3, we observe that the minimum Hamming distance for this cyclic code is dmin = 3. According to Eq. (11.14), the Hamming weight of error patterns that can be corrected is w < {(dmin - 1)/2} = 1, meaning that only single errors can be corrected. As previously discussed, this is a general property of Hamming codes, which is not affected by the choice of cyclic coding. The capabilities of error detection and correction of various cyclic codes are discussed next. Error-correction code types So far, this chapter has provided the basic conceptual tools of ECC principles and coding or decoding algorithms. Here, I shall complete this introduction by review- ing different ECC types used in information technologies (data standards) and telecommunications (packet or frame standards), as well as their error-correction capabiliti