A Class of Binary Signaling Alphabets 



By DAVID SLEPIAN 



(Manuscript received September 27, 1955) 



A class of binary signaling alphabets called "group alphabets" is de- 

 scribed. The alphabets are generalizations of Hamming^ s error correcting 

 codes and possess the following special features: {1) all letters are treated 

 alike in transmission; {2) the encoding is simple to instrument; (3) maxi- 

 mum likelihood detection is relatively simple to instrument; and (4) in 

 certain practical cases there exist no better alphabets. A compilation is given 

 of group alphabets of length equal to or less than 10 binary digits. 



INTRODUCTION 



This paper is concerned with a class of signahng alphabets, called 

 "group alphabets," for use on the symmetric binary channel. The class 

 in question is sufficiently broad to include the error correcting codes of 

 Hamming,^ the Reed-Muller codes," and all "systematic codes''.^ On 

 the other hand, because they constitute a rather small subclass of the 

 class of all binary alphabets, group alphabets possess many important 

 special features of practical interest. 



In particular, (1) all letters of the alphabets are treated alike under 

 transmission; (2) the encoding scheme is particularly simple to instru- 

 ment; (3) the decoder — a maximum likelihood detector — is the best 

 I possible theoretically and is relatively easy to instrument; and (4) in 

 certain cases of practical interest the alphabets are the best possible 

 theoretically. 



It has very recently been proved by Peter Elias^ that there exist group 

 alphabets which signal at a rate arbitarily close to the capacity, C, of 

 the symmetric binary channel with an arbitrarily small probability of 

 error. Elias' demonstration is an existence proof in that it does not 

 show explicitly how to construct a group alphabet signaling at a rate 

 greater than C — e with a probability of error less than 5 for arbitrary 

 positive 5 and e. Unfortunately, in this respect and in many others, our 

 understanding of group alphabets is still fragmentary. 



In Part I, group alphabets are defined along with some related con- 



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