A CLASS OF BINARY SIGNALIHG ALPHABETS 215 



2, • • • , j, «j+i = f some integer, aj+o = ay+s = • • • = «„ = 0, then 

 there does not exist a 2 -letter, w-place alphabet of any sort better than 

 the given (n, A)-alphabet. It will be observed that many of the a's of 

 Table II are of this form. It can be shown that 



Proposition 5 ii n -\- I „ /"t"! q 1^2"^* — 1 there exists 



no 2'''-letter, n-place alphabet better than the best (n, /c) -alphabet. 

 When the inequality of proposition 5 holds the a's are either «o = 1, 



""'' - 1, all other « = 0; or ao = 1, «i = (Vj , «2 = 2"~' - 1 - 



, all other a = 0; or the trivial ao = 1 all other a = which holds 



uhen k = n. The region of the n — k plane for which it is known that 

 (n, A-)-alphabets cannot be excelled by any other is shown in Table IV. 



1.11 A DETAILED EXAMPLE 



As an example of the use of {n, A") -alphabets consider the not un- 

 realistic case of a channel with -p = 0.001, i.e., on the average one binary 

 digit per thousand is received incorrectly. Suppose we wish to transmit 

 messages using 32 different letters. If we encode the letters into the 32 

 5-place binary sequences and transmit these sequences without further 

 encoding, the probability that a received letter be in error is 1 — 

 (1 _ pf = 0.00449. If the best (10, 5)-alphabet as shown in Tables II 

 and III is used, the probability that a letter be wrong is 1 — Qi = 

 1 - r/" - lOgV - 21gy - 24/)' - 72p' + • • • = 0.000024. Thus 

 by reducing the signaling rate by ^^, a more than one hundredfold re- 

 duction in probability of error is accomplished. 



A (10, 5)-alphabet to achieve these results is given in Table III. Let 

 a typical letter of the alphabet be the 10-place sequence of binary digits 

 aia2 ■ • • agttio . The symbols aia^Ozaia^ carry the information and can be 

 any of 32 different arrangements of zeros and ones. The remaining places 

 are determined by 



06 = ai -j- a-i -j- a4 -j- ^5 



a? = tti -j- oo -f a4 -j- as 



as = ai -j- a2 + a.3 + Os 



ag = Oi + 02 4- Qi -j- 0,4 



Oio = Oi + a-i -j- 03 4- 04 4- «5 



To design the detector for this alphabet, it is first necessary to deter- 

 mine the coset leaders for a standard array (4) formed for this alphabet. 



