NON-BINARY ERROR CORRECTION CODES 1353 



values of the corrector, must be at least 2 n + 1, equation (8a). For 

 even bases, we must reject all values of the corrector containing only 

 the digits and b/2 for representing error conditions for the following 

 reasons: a positive error leads to a corrector that is the characteristic of 

 the hicorrectly received digit, and a negative error leads to the ?j-com- 

 plement of such a characteristic. In order to have uniciue error correc- 

 tion, we must be able to distinguish between these two conditions. If a 

 characteristic were to contain only the digits and 6/2, it would be equal 

 to its own 6-complement ; such combinations of digits are therefore not 

 useable as characteristics or characteristic complements. 



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

 digit in case of a single error. 



Convention 1 allows us to distinguish between positive and negative 

 errors. By observing this convention, a characteristic (corresponding to a 

 positive error) can be distinguished from its complement (corresponding 

 to a negative error) by inspecting the first digit of a corrector which is 

 neither nor 6/2. A characteristic will have this digit less than 6/2, 

 a characteristic complement will have this digit greater than b/2. If 

 the corrector is a characteristic, the correction is minus one; if it is a 

 characteristic complement, it is plus one. 



Once the characteristics have been chosen, the corresponding encoding 

 procedure may be performed in the following manner: Let aij represent 

 the Jth digit of the characteristic of information digit .r, . Let Zj represent 

 the check digit which has a characteristic containing a 1 in the jth. 

 position. If convention 2 has been observed, (9) can be used to cal- 

 culate Zj : 



n — m 



J2 (liji'i = —Zj mod 6. (9) 



i=l 



An encoder calculates each Zj and inserts it into the message in those 

 digit positions which have the characteristic of the 7th check digit as- 

 signed to them. 



In more general terms, we use implicit relations that are equivalent 

 to the explicit equations given by (9). Letting x, represent an informa- 

 tion or a check digit, and letting dj represent the jth digit of the charac- 

 teristic of the tth information or check digit, these formulas may be re- 

 written as 



J2 CijXi = mod 6. (10) 



t=i 



At the receiver, the decoder calculates m different check sums. Let Cj 



