A CLASS OF BINARY SIGNALING ALPHABETS 207 



1.3 GROUP ALPHABETS 



An ?i-place group alphabet is a 7v-letter, n-place binary signaling alpha- 

 bet whose letters form a subgroup of Bn . Of necessity the size of an 

 n-place group alphabet is /v = 2 where k is an integer satisfying ^ 

 k ^ n. By an (n, k)-alphahet we shall mean an n-place group alphabet of 

 size 2^. Example: the N{3, 2) = 7 distinct (3, 2)-alphabets are given by 

 the seven columns 



(i) (ii) (iii) (iv) (v) (vi) (vii) 



(3) 



1.4 STANDARD ARRAYS 



Let the letters of a specific (n, /i:)-alphabet be Ai = / = 00 • • • 0, 

 Ao , As , • ■ ■ , A^ , where ju = 2 . The group Bn can be developed accord- 

 ing to this subgroup and its cosets: 



/, A2, A3, ■■• ,A^ 



S2 , S2A2 , S2A3 , • • • , S2A^ 

 Sz , S3A2 , S3A3 , • • • , SsA^ 



Bn = ; (4) 



Sr f SyA2 , SpAz , • ' • , SfAfi 



In this array every element of Bn appears once and only once. The col- 

 lection of elements in any row of this array is called a coset of the (n, k)- 

 alphabet. Here *S2 is any element of B„ not in the first row of the array, 

 S3 is any element of Bn not in the first two rows of the array, etc. The 

 elements S2 , S3 , • • • , Sy appearing under I in such an array will be 

 called the coset leaders. 



If a coset leader is replaced by any element in the coset, the same coset 

 will result. That is to say the two collections of elements 



Si , ^1^2 , SiSz ; ■ • ■ , SiA^ 



and 



SiA,, , (SiAu)A2 , (SiAMs ,■■■ {SiAk)A, 



are the same. 



