annel decoder does, you can use the H at the decoder side in its place. Let's say you send out from the channel coder the 6-bit set umG=(011101. Chapter Six Let's consider Case 1, which is the case when you receive at the channel decoder the 6-bit set V that matches the u sent (i.e., v=u=mG= (011101)). Let's see what happens if we take the received V and multiply it by the parity check matrix H. In this case, we get vH=uH=mGH=m0=0. (6.12) Now let's consider Case 2, the case where the V we receive does not match what we sent from the channel coder, because an error occurred along the way. We'll consider the case where V= (111101). In this case, we can describe what's going on according to V u + e, where u is the sent sequence (011101) and e is the error that occurred, represented as (100000). Let's see what happens when we multiply the received V by the parity check matrix H. In this case we get vH=uH+eH=mGH+eH=m0+eH=0+eH=eH(100000 H (6.13) which is not 0. In fact, doing the math for the parity check matrix of Equation (6.11), you'll find that V H = (100000)H=(100). Let's interpret these results. First, look at Case 1 (no error in received 6-bit set, and V H = 0), then look at Case 2 (an error in received 6-bit set, and vH= (100)). From these two cases, we can determine a very simple result. Any time there is no error, multiplying the received 6-bit vector by H gives 0, and if there is an error, multiplying the received 6-bit error by H does not give 0. So, using H, the channel decoder has an easy way to detect errors. Example 6.5 True or False: the parity check matrix H for the generator matrix (E6.6) (E6.7) Solution: To find out, we simply multiply G H and see if we get zero. Let's find out: Channel Coding and Decoding: Part 1-Block Coding and Decoding (E6.8) Yep. We get zero, confirming that the H of equation (E6.7) is indeed the generator matrix for G in (E6.6). Now let's look into correction of errors. Consider Case 2. We sent u = (011100), the error described by e = (100000) arose, which led to the decoder i