COMPARISON OF SIGNALLING ALPHABETS 



509 



rate in bits per digit at wliicli iut'oiniatioii is being lecieved is 



log K 



R = 



D 



(5) 



In Equation (5) we have neglected a term which takes account of the 

 information lost due to channel noise. This is legitimate because all but 

 10" of the letters are received correctly. 



The worst tolerable probability p of (4) and tlic rate R of Equation 

 (5) determine the noise combating ability of an alphabet. To compare 

 different alphabets one may represent them as points on an efficiency 

 graph of R versus p. Fig. 1 is an efficiency graph on which the values 

 (p, R) for a number of simple error correcting alphabets liave been 

 plotted. Each point on the graph is labelled with the two numbers /;;, D 

 in that order. The alphabets represented were not found by any systema- 

 tic process and are not all proved to be best possible (i.e., to have the 

 largest K) for the stated values of k and D. Fortunately, R depends on 

 K only logarithmically so that it is not likely the points representing the 

 best possible alphabets lie far away from the plotted points. 



The sohd line represents the curve 



i^ = (7 = 1 + p log p + (1 - ?;) log (1 - v)- 



According to Shamion's theorems, all alphabets are represented by 

 points lying below this line. 



The eiSiciency graph only partially orders the alphabets according to 



O 0.4 



0.2 



0.0004 0.001 0.004 0.01 0.02 0.04 0.1 0.2 0.4 0.6 1.0 



PROBABILITY OF ERROR IN ANY DIGIT 



Fig. 1 — Probability of error in a letter is 10~^ 



