NON-BINARY ERROR CORRECTION CODES 1347 



For a single unrestricted error detection code, the complementing 

 digit complements the sum of the information digits. A complementing 

 digit is a check digit. In the binary case, it is a parity check digit. 



As an example, consider a decimal code of this type. A message 823 

 would require a complementing digit 7, making the total message 

 8237 (8 + 2 + 3 + 7 = 20, a multiple of 10). An error in any one digit 

 will mean that the sum of the message digits will not be a nuiltiple of 10. 



For the small error case, it is sufficient to make certain that the sum 

 of all digits is even since any error of ±1 would destroy this property. 

 For the binary case, all errors are small since the only possible error on 

 any digit is a change by ±1; a simple parity check is adequate. For a 

 non-binary code, it would be wasteful to add a digit just to make sure 

 that the sum of all digits is even. In a decimal code for example, if the 

 sum of the message digits is even, the values 0, 2, 4, 6, 8 for the check 

 digit will satisfy a check, or if the sum of the message digits is odd, the 

 values 1, 3, 5, 7, 9 will satisfy the check. More information could l)e 

 sent if a choice among these A'alues could be associated with informa- 

 tion generated by the information source. 



This introduces the concept of a mixed digit ; i.e., a digit which conveys 

 both check information and message information 



A mixed digit is defined as follows: a mixed digit x, base h, is composed 

 of two components (y, z) where y represents an information component 

 and z represents a check component. The number of information states 

 of a mixed digit is jS, wdth y taking the values 0, 1, ■ ■ ■ , — 1 ; the 

 number of check states of a mixed digit is a, the number base of z. 

 In a message containing m check digits and h mixed digits, the number 

 of check states for the message is b"'-ai-a-y ■ . . -au , where a, is the 

 number of check states of the t'th mixed digit. 



If mixed digits are used as part of a code, information must be avail- 

 able in at least two number bases; 5, the number base of the channel, 

 and (3, the number base of the mixed digit. A situation where this aris(»s 

 naturally is in the case of the algebraic sign of a number; this is a digit 

 of information, base 2, which may be associated with other digits of 

 any base. Similarly, any identification which must be associated with 

 numerical information can be conveniently coded in a number base 

 different from the number base of the numerical information. Thus, a 

 mixed digit can sometimes be used conveniently in an information trans- 

 mission system without complicating the infoi'mation sonrco and re- 

 ceptor. 



An error detection code for single small errors suggests tlic use of a 

 mixed digit. In the decimal code for example, the quibinary" representa- 



