Non-Binary Error Correction Codes* 



By WERNER ULRICH 



(Manuscript received April 19, 1957) 



If a noisy channel is used to transmit more than two distinct signals, 

 information may have to be specially coded to permit occasional errors to be 

 corrected. If pulse amplitude modulation is used, the jnost probable error 

 is a small one, e.g., 6 is changed to 7 or 5. Codes for correcting single small 

 errors, and for correcting single small errors and detecting double small 

 errors, in a message of arbitrary length, for an arbitrary number of differ- 

 ent signals in the channel, are derived in this paper. 



For more specialized situations, the error is not necessarily restricted to a 

 small value. Codes are derived for correcting any single unrestricted error 

 in a message of arbitrary length for an arbitrary number of different sig- 

 nals. 



Finally, a set of codes based partially upon the Reed-Muller codes is 

 described for correcting a number of errors in a more restricted class of 

 message lengths for an arbitrary number of different signals. 



The described codes are readily implemented. Many techniques are used 

 which have an analog in a binary system. Other techniques are broadly 

 analogous to binary coding techniques or are special adaptations of a 

 binary code. 



I. INTRODUCTION 



1.1 Use of Error Correction Codes 



One function of an error correction code is to aid in the correct trans- 

 mission of digital information over a noisy channel. This process is 

 illustrated in Fig. 1. An information source gives information to an 

 encoder; the encoder converts the information into a message containing 

 sufficient redundancy to permit the message to be slightly mutilated by 

 the noisy channel and still be correctly interpreted at the destination. 

 The message is then sent ^'ia the noisy channel to a decoder which will 



* This paper was submitted to Columbia University in partial fulfillment of 

 the requirements for the degree of Doctor of Engineering Science in the Faculty 

 of Engineering. 



13-11 



