COMPARISON OF SIGNALLING ALPHABETS .") 1 1 



Hence by takiuj^ D large enough one obtain.s an alphal)et witli rate ex- 

 ceeding Ro — e and letter error probability less than 8. 



The rate Ro appears on the efficiency graph as a dotted line. 



It has not been shown that no error-coirecting alphabet has a rate 

 exceeding Ro . In fact, one alphal)et which exceeds Ro in rate is easy to 

 construct. If the noise probability p is greater than j, then Ro = 0. The 

 alphabet with just two letters 



()()()... 

 and 



1 1 1 1 ... 1 



will certainly transmit information at a (small) positive rate, and with 

 a 10 probability of errors if D is large enough, as long as p < |. 



Using a more refined lower bound for Ko(D, k) it might be shown that 

 there are error-correcting alphabets which signal with rates near C. 

 If one repeats the calculation that led to Ro using the upper bound 

 (3) (which seems to be a better estimate of the true Ko(D, k)) instead 

 of the lower bound (3), one is led to the rate C instead of Ro . 



The condition (4) is more conservative than necessary. The structure 

 of the alphabet may be such that a particular sequence of more than 

 />• errors may occur without causing any error in the final letter. This is 

 illustrated by the following simple example due to Shannon : the alphabet 

 with just two letters 





 111000 



corrects any single error but also corrects certain more serious errors 

 such as receiving 1111 for 0. An alphabet designed for 

 practical use would make efficient enough use of the availat)le sequences 

 so that any seciuence of much more than k errors causes an error in the 

 final letter; the random alphabets constructed above probably do not. 

 If this kind of error were properly accounted for, the rate Ro could be 

 improved, perhaps to C. 



4. Other Discrete Channels 



If instead of transmitting just O's and I's the channel can carry more 

 digits 



0, 1, 2, • . • , n 



