1356 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



If the corrector is 00, the message has been correctly received; other- 

 wise, the corrector is either the characteristic or characteristic comple- 

 ment of the incorrect digit, from which plus one or minus one respec- 

 tively must be subtracted as a correction. 



Consider the general case. Let Xi , x-i , • • • , Xk represent the /.• informa- 

 tion digits; yi , 1/2 , • • • , Vm represent the information state of the m 

 mixed digits, and Zi , 2-2 , • - • , Zm represent the check state of the m 

 mixed digits. In addition, let ai , a2 , ■ • • , am represent the number base 

 of ^1 , 2-2 , • • • , z,„ respectively; I3i , ^2 , • ■ ■ , /3,„ represent the number 

 of possible states of yi , 2/2 , • • • , y,,, respectively, and 0:^+1 , Xk-+2 , ■ • • , 

 Xk+m represent the values of the mixed digits after the message has been 

 encoded. (Note that for simplicity, a check digit is considered as a special 

 case of a mixed digit; its information state is permanently 0.) The follow- 

 ing encoding procedure may be used in which Xi , 2:2 , • • • , Xk are used 

 directly as part of the transmitted message. This is a semi-systematic 

 code, which means that information digits are not changed in coding. 

 To derive the mixed digits, the following formulas are used : 



anXi + • • • + auXk = —zi mod ai (19-1) 



x^k+i) = yioci + zi (20-1) 



ai2Xi + • • • + a2kXk + a2ik+i)X(k+i) = —22 mod 0:2 (19-2) 



X(k+2) = y2a2 + Z2 (20-2) 



ttjiXi + • • • + ajkXk + • • • + aj(k+j-i)X(k+j-i) 



= —Zj mod aj (19-j) 



Z(k+j) = yjaj + Zj (20-j) 



ttmiXi + • • • + a,nkXk + • • • + am(k+m-i)Xa-+m-i) = — z ,n uiod «,„ (19-m) 



X(^k + m) = ymOim + Z^ . (20-m) 



In each case, the value of the check component Zj , of a mixed digit 

 X(k+j) is determined by a formula involving the information digits and 

 previously calculated mixed digits. Immediately after Zj has been de- 

 termined, X(k+j) is calculated for possible use in calculating z^j+d . 

 After the message has been completely encoded the following equations, 

 analogous to (10), will be satisfied. 



