A CLASS OF BINARY SIGNALING ALPHABETS 221 



subgroup of order 2l A fourth independent element can be chosen as 

 any of the remaining 2" — 2 elements, etc. 



Each set of k independent elements serves to generate a subgroup of 

 order 2''. The quantity F{n, k) is not, however, the number of distinct 

 subgroups of C„ of this order, for, a given subgroup can be obtained 

 from many different sets of generators. Indeed, the number of different 

 sets of generators that can generate a given subgroup of order 2^ of C„ 

 is just F{k, k) since any such subgroup is isomorphic with Ck . Therefore 

 the number of subgroups of Cn of order 2'' is N{n, k) = F(n, k)/F(k, k) 

 which is (2). A simple calculation gives N(n, k) = N(n, n — k). 



2.2 PROOF OF PROPOSITIONS 1 AND 2 



After an element A of 5„ has been presented for transmission over 

 a noisy binary channel, an element T of 5„ is produced at the channel 

 output. The element U = AT oi Bn serves as a record of the noise 

 during the transmission. U is an n-place binary sequence with a one at 

 each place altered in A by the noise. The channel output, T, is obtained 

 from the input A by multiplication by U: T = UA. For channels of the 

 sort under consideration here, the probability that U be any particular 

 element of Bn of w^eight w is p^'g"""'. 



Consider now signaling with a particular (n, /b) -alphabet and consider 

 the standard array (4) of the alphabet. If the detection scheme (8) is 

 used, a transmitted letter A i will be produced without error if and only 

 if the received symbol is of the form SjAi . That is, there will be no 

 error only if the noise in the channel during the transmission of Ai is 

 represented by one of the coset leaders. (This applies (or i = 1,2, • • • , 

 fi = 2 ). The probability of this event is Qi (Proposition 1, Section 1.6). 

 The convention (5) makes Qi as large as is possible for the given alpha- 

 bet. 



Let X refer to transmitted letters and let Y refer to letters produced 

 by the detector. We use a vertical bar to denote conditions when writing 

 probabilities. The quantity to the right of the bar is the condition. We 

 suppose the letters of the alphabet to be chosen independently with 

 ec^ual probability 2" . 



The equivocation h{X \ Y) obtained when using an (n, fc)-alphabet 

 with the detector (8) can most easily be computed from the formula 



h(X I F) = h{X) - h(Y) + h(Y I X) (10) 



The entropy of the source is /i(X) = k/n bits per symbol. The probability 

 that the detector produce Aj when Ai was sent is the probability that 

 the noise be represented by AiAjSt , ^ = 1,2, • • • , v. In symbols, 



