138-4 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



limit to the efficiency of codes may be found. This Umit can be derived 

 from the fact that an error correction code decoder and correction cir- 

 cuit must be able to convert any message which contains errors within 

 the bounds of the correction performed by the code, into the value of 

 the message as originally transmitted, or must be able to derive the 

 original information which was fed into the encoder. Thus, if each mes- 

 sage may be mutilated in w ways, and still be corrected, then at least w 

 messages must be associated with each allowed message. This is indi- 

 cated diagrammatically in Fig. 4. The messages produced by the encoder 

 are shown at the left; each one fans out to w — 1 mutilated messages 

 plus the original message. The decoder converts any of these w messages 

 into the original message. 



The value of w can be determined by taking all possible combinations 

 of errors that can be corrected by a coding system. For example, for a 

 code system which can correct up to {d — l)/2 small errors in different 

 digits in an n digit message, w is given by 



w 



(d-l)/2 



(55) 



i=0 



where d is the minimum distance between messages, and 





n! 



(n — i)li\ 



This equation merely signifies that w is the sum of all combinations of 

 positive and negative (accounting for the 2 term) errors in up to 

 {d — l)/2 different digits out of n digits. For single errors, iv = 2n -f 1. 



INFORMATION 

 SOURCE 



ENCODER 



CHANNEL 



DECODER 



INFORMATION 

 RECEPTOR 



W DIFFERENT 

 ^7 MESSAGES 



Fig. 4 — Graphicjil representation of an error correction code. 



