208 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G 



We define the weight Wi = w{Ti) of an element, Ti , of Bn to be the 

 number of ones in the n-place binary sequence T,- . 



Henceforth, unless otherwise stated, we agree in dealing with an ar- 

 ray such as (4) to adopt the following convention: 



the leader of each coset shall be taken to be an . . 



element of minimal weight in that coset. 



Such a table will be called a standard array. 



Example: Bi can be developed according to the (4, 2)-alphabet 0000, 

 1100, 0011, nil as follows 



(6) 



(7) 



The coset leader of the second coset of (6) can be taken as any element 

 of that row since all are of weight 2. The leader of the third coset, how- 

 ever, should be either 0010 or 0001 since these are of weight one. The 

 leader of the fourth coset should be either 1000 or 0100. 



1.5 THE DETECTION SCHEME 



Consider now communicating with an (n, fc) -alphabet over the sym- 

 metric binary channel. When any letter, say A,, of the alphabet is 

 transmitted, the received sequence can be of any element of B„ . We 

 agree to use the following detector: 



if the received element of Bn lies in column i of the array (4), the 



detector prints the letter Ai ,i = 1,2, • • • , ju. The array (4) is to (8) 



be constructed according to the convention (5). 



The following propositions and theorems can be proved concerning 

 signaling with an (n, /c)-alphabet and the detection scheme given by (8). 



1.6 BEST DETECTOR AND SYMMETRIC SIGNALING 



Define the probability /,• = ((Ti) of an element Ti of Bn to be A = 

 ^wi^n-uf ^yYiere p and q are as in (1) and Wi is the weight of Ti . Let 



