204 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



cepts necessary for their understanding. The main results obtained up 

 to the present time are stated without proof. Examples of these concepts 

 are given and a compilation of the best group alphabets of small size 

 is presented and explained. This section is intended for the casual reader. 



In Part II, proofs of the statements of Part I are given along with 

 such theory as is needed for these proofs. 



The reader is assumed to be familiar with the paper of Hamming, 

 the basic papers of Shannon* and the most elementary notions of the 

 theory of finite groups. 



Part I — Group Alphabets and Their Properties 



1.1 INTRODUCTION 



We shall be concerned in all that follows with communication over the 

 symmetric binary channel shown on Fig. 1. The channel can accept 

 either of the two symbols or 1 . A transmitted is received as a with 

 probability q and is received as a 1 w'ith probability p — 1 — g : a trans- 

 mitted 1 is received as a 1 with probability q and is received as a with 

 probability p. We assume ^ p ^ ^^. The "noise" on the channel 

 operates independently on each symbol presented for transmission. The 

 capacity of this channel is 



C = 1 + P log2P + q log29 bits/symbol (1) 



By a K-leUer, n-place binary signaling alphabet we shall mean a collec- 

 tion of K distinct sequences of n binary digits. An individual sequence 

 of the collection will be referred to as a letter of the alphabet. The integer 

 K is called the size of the alphabet. A letter is transmitted over the 

 channel by presenting in order to the channel input the sequence of n 

 zeros and ones that comprise the letter. A detection scheme or detector for 



INPUT X OUTPUT 



Fig. 1 — The symmetric binary channel. 



