1341) THE BELL SYSTEM TECHNICAL JOUKXAL, NOVEMBER 1957 



eral solution has been found for multiple error correction codes for the 

 unrestricted error case. 



In Section VI, a number of techniques are presented for using binary 

 error correction coding schemes for non-binary error correction codes. 

 Section 6.1 shows how such techniques may be used to obtain non-binary 

 single error correction codes, and single error correction double error 

 detection codes, for the small error case. Section 6.2 presents a special 

 technique, involving the use of an adaptation of the Reed-Muller binary 

 code, to obtain a class of non-binary multiple error correction codes, for 

 the small error case. 



Section VII shows that an iterative technique of binary coding can be 

 directly applied to non-binary codes. It also shows how an adapted 

 Reed-Muller code can be profitably used in such a system. 



Section VIII summarizes the results obtained in Sections II-VII and 

 shows the advantages and shortcomings of many of these codes. 



Section IX presents general conclusions which may be drawn from 

 this paper. 



II. SINGLE ERROR DETECTION CODES 



Single error detection codes rec^uire message points separated in n 

 dimensional space by a distance of two. 



For the binary case, the only two possible types of errors are the 

 change from a 1 to a and from a to a 1. 



A simple technique that is used frequently for binary error detection 

 codes is to encode all messages in such a manner that every message 

 contains an even number of I's. This is accomplished by adding a 'parity 

 check digit to the information digits of a message; this digit is a 1 if an 

 odd number of I's exist in the information digits of a message and is a 

 if an even number of I's exist in the information digits. At least two 

 errors must occur before a message containing an even number of I's 

 can be converted into another message containing an even number of 

 I's, since the first error will always cause an odd number of I's to 

 appear. A message with an odd number of I's is known to be incorrect.* 



An analogous technique may be used for the unrestricted error case in 

 non-binary codes. We can obtain a satisfactory code by adding a com- 

 plementing digit to a series of information digits to form a message. 



A complementing digit, base b, is defined as a digit which when added 

 to some other digit will yield a multiple of 6. 



* Parity check digits may be selected to make the number of I's in a message 

 always odd, but the principle is the same; in this case, an error is recognized if a 

 received message contains an even number of I's. 



