NON-BINARY ERROR CORRECTION CODES 1371 



number less than a,„/2 in its last position. The rest of its digits are 

 arbitrary. 



Convention 5. The above conventions restrict the choice of charac- 

 teristics. In order to have n distinct characteristics, m mixed or check 

 digits, using check bases ai , a-, , •■-,««, are required, and inec{uality 

 (51) must be satisfied: 



n ^ aia2- ... -am-i-g. (51) 



Codes may be derived using the above conventions only if b ^ 4. 

 For the ternary case, a relatively efficient code may be obtained by 

 using one ternary digit as an over-all parity check digit. The rest of the 

 message is in a single small error correction code, derived using the 

 rules and conventions of Section III. Any single small error will lead to a 

 failure of the parity check, and a double small error will lead to a failure 

 of other checks but not the parity check. 



No general solution has been found for deriving an efficient single 

 error correction double error detection code for the unrestricted error 

 case. Also, no general solution has been found for deriving an efficient 

 multiple error correction code for the unrestricted error case. A reason- 

 ably efficient method has been found for correcting multiple errors in 

 the more important small error case; this is discussed in Section 6.2. 



VI. THE USE OF BINARY ERROR CORRECTION TECHNIQUES IN NON-BINARY 

 SYSTEMS 



In this section, methods for using binary codes for the correction of 

 errors in a non-binary system are described. Although the single small 

 error correction codes obtained in this manner are generally less flexible 

 than the codes obtained in Section III, the class of multiple error correc- 

 tion codes described in Section 6.2 is the only reasonably satisfactory 

 class of such codes that has been found. The codes described in this 

 section are semi-systematic but are not simple characteristic codes. 



6.1 Single Small Error Correction Codes 



Binary codes are most conveniently used for correcting small errors 

 (±1). Suppose any digit, base h, has an associated pair of binary digits, 

 arranged in such a way that a change of ±1 in the base b digit will 

 change only one of the two binary digits. For b = 10, an association 

 such as the one shown in Table IX might be used. For example, if a 

 6 is received as a 7, the associated binary message would indicate 

 that the second of the binary digits is incorrect; a 7 can be corrected 



