A CLASS OF BINARY SIGNALING ALPHABETS 209 



Qi , i = 1 , 2, • • • , jLi be the sum of the probabilities of the elements in 

 the iih. column of the standard array (4). 



Proposition 1. The probability that any transmitted letter of the 

 (n, A;) -alphabet be produced correctly by the detector is Qi . 



Proposition 2. The equivocation^ per symbol is 



1 ** 

 Hy{x) = — S Qi log2 Qi 



n i=i 



Theorem 1 . The detector (8) is a maximum likelihood detector. That 

 is, for the given alphabet no other detection scheme has a greater average 

 probability that a transmitted letter be produced correctly by the de- 

 tector. 



Let us return to the geometrical picture of w-place binary sequences 

 as vertices of a unit cube in n-space. The choice of a i^-letter, n-place 

 alphabet corresponds to designating K particular vertices as letters. 

 Since the binary sequence corresponding to any vertex can be produced 

 by the channel output, any detector must consist of a set of rules that 

 associates various vertices of the cube with the vertices designated as 

 letters of the alphabet. We assume that every vertex is associated with 

 some letter. The vertices of the cube are divided then into disjoint sets, 

 Wi , Wi , • • • , Wk where Wi is the set of vertices associated with tth 

 letter of the signaling alphabet. A maximum likelihood detector is char- 

 acterized by the fact that every vertex in Wi is as close to or closer to 

 the iih. letter than to any other letter, i = 1,2, • • • , K. For group alpha- 

 bets and the detector (8), this means that no element in the iih. column 

 of array (4) is closer to any other A than it is to ^i , z = 1, 2, • • • , ;u. 



Theorem 2. Associated with each {n, /(;)-alphabet considered as a point 

 configuration in Euclidean n-space, there is a group of n X n orthogonal 

 matrices which is transitive on the letters of the alphabet and which 

 leaves the unit cube invariant. The maximum likelihood sets 1^1 , 

 W2 , • • • Wn are all geometrically similar. 



Stated in loose terms, this theorem asserts that in an (n, A;)-alphabet 

 every letter is treated the same. Every two letters have the same number 

 of nearest neighbors associated with them, the same number of next 

 nearest neighbors, etc. The disposition of points in any two W regions 

 is the same. 



1.7 GROUP ALPHABETS AND PARITY CHECKS 



Theorem 3. Every group alphabet is a systematic^ code: every syste- 

 matic code is a group alphabet. 



