NON-BINARY ERROR CORRECTION CODES 1357 



Let Cij represent aji in equation (19-j). Then, 



k-\-w, 



X CijXi = mod a,. (21) 



1=1 



(Since Xqc+j) = ^j rood ay, substitution of X(k+j) ioY Zj inequation (19-j) 

 will continue to satisfy the equation.) 



At the decoder, equation (21) is changed to 



k-\-m 



^ CijXi = Cj mod aj (22) 



i=l 



In (22), Xi represents the received value of Xi , and Cj represents the jth 

 digit of the corrector. If all the digits have been correctly received, i.e., 

 x/ = Xi for all values of i, then Ci = C2 = • • • = c,„ = 0; [see equation 

 (21)]. If Xh. had been received incorrectly so that xi/ = xi, + 1, but all 

 other digits had been correctly received, then the value of Cj (the ^th 

 digit of the corrector) would be calculated in the following manner: 



k+m 



Cj mod aj = ^ CijXi' 



k-^m 



Cj mod aj = Yl CijXi + Chj = Chj (23) 



1=1 



Equation (23) proves that Chj is actually the Jth digit of the charac- 

 teristic of Xh , because by definition, the characteristic of Xh is the value 

 of the corrector when Xh = Xh -{- 1 , and all other digits have been cor- 

 rectly received. This means that the general term. C,> of (21), is actually 

 the jih digit of the characteristic of the ith. digit and that this is a simple 

 characteristic code. 



For the case that Xh = Xh — 1, the value of the corrector is such that 

 if it were incremented, digit by digit, by the characteristic of Xh , the 

 corrector would be composed only of zeros. Incrementing the corrector 

 by the characteristic of Xh is equivalent to recalculating the corrector 

 with .T;,' increased by one, which in this case would amount to calculat- 

 ing the corrector for the case of a correctly received message. The 

 latter is composed of all zeros [see (21)]. Thus, for the case of a single 

 error of —1, the corrector is the characteristic complement of the digit 

 which is incorrectly received. For a semi-systematic or systematic code, 

 the characteristic complement is an m digit word whose Jth digit is the 

 complement modulo aj of the jth digit of the characteristic. 



E(iuation (20-j) shows that generally aj^j cannot exceed b. (An ex- 

 ception is given below.) The maximum \'alue of i/j is /3> — 1 since y is a, 



