A CLASS OF BINARY SIGNALING ALPHABETS 225 



parity check sequences. It follows then that all elements in a given coset 

 have the same parity check sequence. For, let the coset be Si , SiA2 , 

 SiAz , ■ ■ • SiA^ . Since the elements I, A^ , A3, • • • , A^ all have parity 

 check sequence 00 • • • 0, all elements of the coset have parity check 

 R(Si). 



In the array (4) there are 2" cosets. We observe that there are 2"~* 

 elements of Bn that have zeros in their first k places. These elements 

 have parity check symbols identical with the last n — k places of their 

 symbols. These elements therefore give rise to 2"~ different parity check 

 symbols. The elements must be distributed one per coset. This proves 

 Theorem 4. 



2.7 PROOF OF PROPOSITION 5 



If 



n ^ 2"-' - 



we can explicity exhibit group alphabets having the property mentioned 

 in the paragraph preceding Proposition 5. The notation of the demon- 

 stration is cumbersome, but the idea is relatively simple. 



We shall use the notation of paragraph 2.1 for elements of Bn , i.e., 

 an element of Bn will be given by a list of integers that specify what 

 places of the sequence for the element contain ones. It will be convenient 

 furthermore to designate the first k places of a sequence by the integers 

 1, 2, 3, • • • , k and the remaining n — k places by the "integers" 1', 2', 

 3', • • • , r, where ( = n — k. For example, if n = 8, /c = 5, we have 



10111010^ 13452' 

 10000100^ 11' 

 00000101 ^ 1'3' 



Consider the group generated by the elements 1', 2', 3', • • • , (' , i.e. 

 the 2' elements /, 1', 2', ■■■,(', 1'2', 1'3', • • • , 1'2'3' ■■■('. Suppose 

 these elements listed according to decreasing weight (say in decreasing 

 order when regarded as numbers in the decimal system) and numbered 

 consecutively. Let Bt be the zth element in the list. Example: if ( ^ 3, 

 Ih = 1'2'3', B2 = 2'3', B, = 1'3', B, = 1'2', B, - 3', B, = 2', B, - 1'. 



Consider now the (n, /^-alphabet whose generators are 



ISi , 2B, ,W,, ■■• , kBk 

 We assert that if 



