1344 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



used in this paper. A small error changes a digit to an adjacent value. 

 In a decimal system, a change from 1 to 2 or 1 to is a small error. An 

 unrestricted error changes a digit to any other value. In a decimal sys- 

 tem, a change from 1 to 5 is an unrestricted error. 



1..3 Material To Be Presented 



The various types of codes described in this paper and the sections 

 in which they are to be found are summarized in Table 1. The tech- 

 niques which are described are summarized below. 



The geometric model suggests the simplest approach to error correction 

 codes. A transmitter has a "codebook" containing all members of the 

 set of transmitted messages. If the message source gives to the encoder 

 the signal that the information to be sent is k (that is to say, the A'th 



output of all the outputs associated with the message source), the en- 

 coder chooses the kth member of the set. The decoder will then look up 

 the message it receives in its own codebook which contains all possible 

 received messages, and corresponding to the entry of the received mes- 

 sage will find the symbols corresponding to k. Or the receiver may 

 compare the received message with every member of the set of trans- 

 mitted messages, calculate the distance between the two, and correct 

 the received message to whichever of the transmitted messages is sep- 

 arated from the received message by the smallest distance. (It has been 

 shown by Slepian* that this is the message most likely to be correct in a 

 symmetrical binary channel having the property that changes from 1 to 

 and from to 1 as a result of noise in the channel are equally likely.) 

 The practical difficult}' with such a code is the large size of the re- 

 quired codebooks. Most coding schemes try to eliminate such codebooks 

 and substitute a set of rules for encoding, decoding and correcting 

 messages. 



