NON-BINARY ERROR CORRECTION CODES 



1355 



Table IV — Characteristics and Characteristic Complements, 

 Semi-Systematic Decimal Code 



five corrector states are required, one for each of the two possible cor- 

 rections (±1) for each digit, and one for the case of a correctly received 

 message. Characteristics may be chosen for the various digits in accord- 

 ance with the rules and conventions outlined above in this case, since 

 the check modulus is the same for both check digits. Consequently, it 

 is no accident that these characteristics, shown in Table IV, are the 

 same as those shown in Table III. 



Let Ca and Ci2 represent the characteristic of the ith digit, and let 

 l/i and 2/2 represent the two binary information digits. Then: 



12 



£ CaXi ^ —zi mod 5, Xi = 21 + 5y, 



i=3 

 12 



^ Ci2Xi = —Z2 mod 5, 0^2 = 22 + 5yo . 



(13) 



(14) 



4 = 3 



Because Xi = Zi mod 5 and X2 == 22 mod 5, these relations can be re- 

 written implicity to resemble equation (10): 



12 



23 CnXi = mod 5, 



i=l 

 12 



^ Ci2Xi = mod 5. 

 1=1 



At the decoder, the corrector C1C2 is calculated by: 



12 



S C'tl•^•^•' = Ci mod 5, 

 1=1 



12 



^ Ci2Xi' = C2 mod 5. 



(15) 

 (16) 



(17) 

 (18) 



t=i 



