it errors and we want them to correct bit errors. So, we will measure the performance of block coders by determining how well they are able to do both. Specifically, we'll make our performance measures: 1. Pm, m' the probability that a channel decoder failed to detect (or missed) a bit error; and/or 2. P, the probability that a channel decoder failed to correct a bit error. With our measuring stick in hand, let's revisit the channel coders we originally looked at, and determine Pmand/or P. 6.3.1 Performances of Single Parity Check Bit Coders/Decoders For single parity check bit coders/decoders, we receive n = (k + 1) bits and can always detect an odd number of errors in these bits. But we always fail to detect an even Channel Coding and Decoding: Part 1-Block Coding and Decoding number of errors. So, the probability Pm (the probability we missed an error) is just the probability that an even number of errors occurred. That is, (6.16) jeeven where P(j,n) refers to the likelihood of having j bit errors occur in a block of n bits. This value can be found from some statistics literature, but rather than make you look it up, I'll just tell you that it refers to (6.17) where 1. (")res refers to the value j!(n-j): and 2. p is the probability that a bit error occurs when a bit travels from channel coder output to channel decoder input (i.e., the probability that a bit error occurs when a bit is sent through the modulator, across the channel, and finally through the demodulator-look at Figures 6.1 and 6.2 to picture this). Also, since these single parity check bits can't correct any errors, we know that P=1 = (i.e., you're 100% sure that you won't correct an error). 6.3.2 The Performance of Rectangular Codes We saw that rectangular codes map k = M.N bits to n = (M+1) (N+1) bits, and by doing this they can correct one bit error. One thing I didn't show you earlier is that the rectangular codes cannot correct two or more errors, just that one. (You can show this to yourself by putting two errors in the rectangular code and 