ror you think happened. Here's an example to illustrate the point. Consider the channel coder of Figure 6.10, which has G as shown in Equation (6.2) and H as given in Equation (6.11). Following step 1, you can create a table linking possible error vectors e to S = e H. When you do this, you'll get results matching Figure 6.11. Now, let's say you sent u= (101110), and you received V=u+e=u+ (100000)= (001110). Computing V H, we get vH=eH=S=(100). (6.15) Now looking to Figure 6.11, we see this syndrome corresponds to error e= (100000), SO we correct for this error. With the error corrected, we then figure out that the 3-bit input was simply (110) using Figure 6.10. Good work! Example 6.6 Determine a table of errors and corresponding syndromes for the 2-bit to 4-bit linear block coder described in Table E6.2 and described by generator matrix 1110 1 (E6.9) Channel Coding and Decoding: Part 1-Block Coding and Decoding Table E6.2 Linear block code and with parity check matrix (E6.10) Solution: 1. The first error to consider is the no-error case of e = (0000). In this case, we have (E6.11) 2. The second cases to consider are the one-bit errors, starting with e = (0001), in which case we have the syndrome (E6.12) 3. Next, we'll consider the one-bit error = (0010), in which case the syndrome corresponds to (E6.13) Chapter Six 4. Continuing on, we consider the one bit error ((0100), in which case Se(100)=(0 1 0 0) (E6.14) 5. That's it. We're out of syndromes. So, our table of errors to syndromes corre- sponds to Table E6.3. So, whenever we receive a set of four bits V, we multiply it by H, and we get v H = e H = S. We then use the syndrome table to determine which error e that corresponds to, and correct that error. Table E6.3 Mapping syndromes and errors 6.3 Performance of the Block Coders Now that you know how these blocks coders work, we'll characterize their perfor- mance in order to tell which block coders are better than others. We'll first think-hmmm, what do we want these block coders to do? We want them to detect b