A CLASS OF BINARY SIGNALING ALPHABETS 



205 



a given /v-letter, n-place alphabet is a procedure for producing a sequence 

 of letters of the alphabet from the channel output. 



Throughout this paper we shall assume that signaling is accomplished 

 with a given /i-letter, n-place alphabet by choosing the letters of the 

 alphabet for transmission independently with equal probability l/K. 



Shannon^ has shown that for sufficiently large n, there exist K-letter, 

 n-place alphabets and detection schemes that signal over the symmetric 

 binary chaimel at a rate R > C — e for arbitrary £ > and such that 

 the probability of error in the letters of the detector output is less than 

 any 5 > 0. Here C is given by (1) and is shown as a function of p in 

 Fig. 2. No algorithm is known (other than exhaustvie procedures) for 

 the construction of A'-letter, /i-place alphabets satisfying the above 

 inequalities for arbitrary positive 8 and e except in the trivial cases C — 

 and C = 1. 



1.2 THE GROUP -S„ 



There are a totality of 2" different w-place binary sequences. It is fre- 

 quently convenient to consider these sequences as the vertices of a cube 

 of unit edge in a Euclidean space of n-dimensions. For example the 5- 

 place sequence 0, 1, 0, 0, 1 is associated with the point in 5-space whose 



o.e 



0.6 



0.4 



0.2 



Fig. 2 — The capacity of the symmetric binary channel. 

 C = 1 + p log2 p + {I - p) log2 (1 - p) 



