r sum is O Does not detect 2 errors bit in bit in error error Figure 6.4 Error detection at channel decoder (a) sent bits (b) 1 bit received in error (c) 2 bits received in error Chapter Six Let's consider whether or not we're able to correct (and not just detect) any trans- mission errors using single parity check bits. Look at Figure 6.5(a), to see the sent bits. In Figure 6.5(b) and 6.5(c), you see two cases of received bits, each with a different bit in error. Now, compute the sum for the case of Figure 6.5(b) and the case of Figure 6.5(c)-in both cases we get a sum of 1. This tells us there's an error, but we have no way of knowing which bit is in error-we can't tell if it's the case of Figure 6.5(b) or Figure 6.5(c). Therefore, single parity check bits cannot be used to correct any errors. sum is 1 ERROR! bit in error sum is 1 ERROR! bit in error Figure 6.5 The lack of error correction at the channel decoder (a) sent bits (b) 1 bit received in error (Case 1) (c) 1 bit received in error (Case 2) Example 6.1 Determine the output of a parity check bit coder which, for every 3 bits in, puts 4 bits out. Assume the input is 001 110. If bit errors occurred in the first two bits, could a decoder detect this? Solution: With input 001 110, a new bit is added after each 3 bits. That bit makes sure that the sum of each set is now 0. So, with 110 coming in, 001 1 110 0 comes out. A 1 is added after the first three bits, with a 0 added after the final three bits. If the first two bits are in error, we receive 1111110 0. At the receiver, we form a sum over each set of 4 bits and make sure the sum adds to 0. If it does, we say "no error detected"; if it doesn't add to 0, we say "error detected." So in this case, we create 1+1+1+1 (modulo 2 sum of first 4 bits) 0, and over the next set of 4 bits we create 1+1+0+0 (modulo 2) = 0. So, even though two errors occurred, none are detected. Channel Coding and Decoding: Part 1-Block Coding and Decoding 6.1.2 Some Terminology Before we move on to consider the more sophisticated form