508 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



Table I 

 Table of 2^/N{D, k) 



Missing entries are numbers between 1 and 2. 



This sort of criterion might be appropriate for a channel transmitting 

 English text. For other messages it is not always appropriate. For ex- 

 ample, if the messages are telephone numbers, one would naturally 

 require that the probability of mistaking a telephone number be small, 

 say less than 10 . If the telephone numbers are L decimal digits long, 

 and if the alphabet has K different letters in it (so that it takes about 

 L log 10/log K letters to make up a telephone number) the probability 

 of making a mistake in a single letter should be required to be less than 

 about 



10"' log K 

 L log 10 



which gives alphabets with large K an advantage over alphabets with 

 small K. 



Since the probability that exactly r binary digits out of D shall be 

 received incorrectly is Cu^rV" (1 ~ p) ^ we achieve the required re- 

 liability with a D-dimensional fc-error correcting alphabet provided p 

 satisfies 



T=k+\ 



(4) 



The value of p which makes the inequality hold with the equals sign 

 determines the noisiest channel over which the alphabet can be used 

 safely. 



Let K be the number of different letters in the alphabet. Then the 



