1354 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



represent the check sum corresponding to the jth corrector term, and 

 Xi represent the received vahie of Xi : Then, 



n 



^ CijX/ = Cj mod b. (11) 



t=i 



The difference between equations (10) and (11) is the result of any 

 mutilations caused by the channel. If no error has occurred, all the c/s 

 are 0; if an error of ±1 has occurred, the m c/s will form the characteristic 

 or the characteristic complement, respectively, of the incorrectly re- 

 ceived digit. 



One disadvantage of a systematic code is the discontinuity in the 

 number of check states as a function of w, the number of check digits. 

 For example, in decimal code one check digit is required for a message 

 of up to four digits, and two check digits for up to forty-eight digits. 

 Obviously, for a message of intermediate length, for example, twelve 

 digits, many of the corrector states cannot be used for single error cor- 

 rection smce they will not correspond to any single error. A more effi- 

 cient code would be obtained if the check states were limited to a smaller 

 number. 



One method of reducing the number of check states is to perform the 

 check in a different modulus than the modulus of the channel. In the 

 single error detection code using a mixed digit, binary check informa- 

 tion and quinary message information was conveyed by this digit. This | 

 code was more efficient than a systematic code because each message 

 contained the minimum number of check states which is 2. 



If a mixed digit, x, is composed of the two components (y, z) where y 

 is the information state of the digit and z the check state, it is conven- 

 ient to combine these two components to form x by means of the formula , 



x = ay -\- z. (12) 



We calculate z by using a linear congruence equation modulo a. 



The use of this formula permits a decoder to act on x', the received 

 value of x, directly, without first resolving x' into y' and z', because (12) 

 insures that x' = y' mod a. This permits x' to be corrected directly and 

 then resolved into its components. 



As an example, consider a semi-systematic code for correcting a single 

 small error in a decimal system, using a twelve digit message; ten of the 

 digits are information digits and two are mixed digits, each conveying 

 binary message information and quinary check information. (One of 

 these binary digits might represent the sign of the number.) 



With two quinary checks, twenty-five different check states are pos- 

 sible ; for correcting single small errors in a twelve digit message, twenty- 



