COMPARISON OF SIGNALLING ALPHABETS 507 



k error correcting alphabet can contain. Except wlien k = I, there i.s no 

 general method for constructing an alphabet with /vo(I>, k) letters, nor 

 is A'o(D, />•) known as a function of 1) and A'. Crude upper and lower 

 bounds for Ki,(D, k) are given by the following theorem. 

 Theorem 1. The largest number of letters 7v ()(/->, k) satisfies 



where 



N{D, k) = T.Co,r 



r=0 



is the number of sequences of D digils which differ from a given sequence 

 in 0, 1, ' ■ • , or k places. 



Proof 



The upper liound is due to R. W. Hamming and is proved by noting 

 that foi' each letter S of a A" error correcting alphabet there are N(D, k) 

 possible received sequences which will be interpreted as meaning S. 

 Hence N(D, k) Ko(D, k) < 2 , the total number of sefiuences. 



The lower bound is proved by a random construction method. Pick 

 any sequence Si for the first letter. There remain 2 '' — N(D, 2k) se- 

 (luences which differ from »S'i in 2A- + 1 or more places. Pick any one of 

 these 8-2 for the second letter. There remain at least 2 — 2N{D, 2k) 

 sequences which differ from both Si and S-> in 2A' + 1 or more places. 

 As the process is continued, there remain at least 2 — rN(D, 2k) 

 secjuences, which differ in 2A- + 1 or more places from *S'i , • • • , Sr , 

 from which *SV+i is chosen. If there are no choices available after choosing 

 Sk , then 2'' - KN{D, 2k) < so the alphabet (Si , ■ ■ • , Sk) has at 

 least as many letters as the lower bound (3). 



For all the simple cases (D and A- not very large) investigated so far 

 the upper liound is a better estimate of Ko(D, k) than the lower bound. 

 The upper and lower bounds differ greatly, as may be seen from a quick 

 insjx'ctioii of Table I. For example, in the case of a ten dimensional two 

 error correcting alphabet, the bounds are 2.7 and 18.3. 



2. Efficiency Graph 



The first step in constructing an efficiency graph for comparing alpha- 

 bets is to decide on what constitutes reliable transmission. The criterion 

 used here is that on the average no more than one lettei- in 10 shall be 

 misinterpreted. 



