NON-BINARY EIIKOR COKKECTION CODES 1359 



characteristic of the mth mixed digit must therefore contain only a single 

 digit which is not 0, a,„ must be greater than 2. 



Convention 1 . The various digits of a characteristic are arranged in a 

 set order, i.e., Ca , Ca, • • • , dm • The first digit which is neither nor 

 (Xj/2 must be less than olj/2. There must be at least one such digit. 



Convention 2. The characteristic of the jth mixed digit has a 1 in the 

 jth position and O's elsewhere, provided that ay 9^ 2. If a^ = 2, the char- 

 acteristic of this mixed digit has a 1 in the jth and mth positions, and O's 

 elsewhere. 



Rule 1 is required because the number of possible corrector states is 

 ai-ar ... •oim , of which only those containing at least one digit which 

 is neither nor a/2 can be associated with the 2n possible errors. The 

 same reasons used for Rule 1 for the systematic code case are equally 

 applicable here; a characteristic containing only the digits or a:y/2 in 

 the jth position is not distinguishable from its complement. 



Rule 2 is required to permit a unique identification of an incorrect 

 digit. 



Rule 3 is necessary to derive the sign of an error on the mth mixed digit. 



The reasons for using Conventions 1 and 2 in the case of the sys- 

 tematic code are equally applicable in this case. For the case a = 2, 

 however, a special convention must be used to avoid a conflict with 

 Convention 1. 



The procedure for converting a set of characteristics into an error 

 correcting code system is the same for a semi-systematic code as for a 

 systematic code except that the following additional functions must be 

 performed: the encoder must combine check states with information 

 states to derive mixed digits, and the decoder must resolve mixed digits 

 into information and check digits offer it has performed its corrections. 



By using these rules and conventions, the most efficient simple charac- 

 teristic code can be determined. For messages of length n (including 

 mixed or check digits), the following relations must be satisfied: 



Let 



P = avoi-r ... •«,„ , 



Q = ^v^-r ... ■^,n, 



mi = number of even as. 



Then: 



(P - 2"'0/2 ^ n, (28) 



a^^i ^ b. (29)* 



For exceptions, see above. 



