 or bandwidth waste. Note that in another possible convention for the systematic form/code, as described in some textbooks, the parity bits may instead precede the payload bits. Error correction For future use, we define the parity-check matrix: = [PT|Im]= (11.4) where is the transposed matrix of P (note how their coefficients are symmetrically permuted about the diagonal elements). We note then the important property HGT==HH== 0, (11.5) which stems from (11.6) In this expression, we have used the property that for binary numbers X, the sum x x x is identical to zero (0 ++0=1+1=0). From the property in Eq. (11.5), we have YHT=0, (11.7) which stems fromYHX(GHT)= = 0. We consider now the received block code, which we can define as the n-vector Z. Since the received block code is a "contaminated" version of the original block code Y, we can write Z in the form: Z=Y+E, (11.8) where E is an error vector whose ith coordinate is 0 if there is no error and 1 otherwise. Post-multiplying Eq. (11.8) by T and using the property in Eq. (11.7), we obtain (11.9) The m-vector S is called the syndrome. The term "syndrome" is used to designate the information helping to diagnose a "disease," which, here, is the error "contamination" of the signal. A zero-syndrome vector means that the block contains no errors (E = 0). As the example described next illustrates, single-error occurrences (E has only one nonzero coordinate) are in one-to-one correspondence with a given syndrome S. Thus when computing the syndrome S = ZH™ at the receiver end, one immediately knows two things: (a) Whether there are errors, as indicated by S # 0 ; (b) In the assumption that it is a single-error occurrence, where it is located. At this point, we should illustrate the process of error detection and error correction through the following basic example. Consider the block code (n,k) = (7,4 4), which has k = 4 message bits and m = 3 parity bits. Such a block, which is of the form n = 2m - 1 and k=n- - m, with m > 3, 11.2 Linear block codes Table 11.1 Message 