1362 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



rection and one for the correct message. When m check digits are used, 

 p™ corrector states are obtained. 



Rule 2 and Convention 2 are the same for the single small error cor- 

 rection systematic codes. The same reasons apply for both cases. 



Convention 1 is changed from the equivalent convention for the small 

 error correction code, because the magnitude of the error, not only its 

 sign, must be derivable for a code for correcting single unrestricted 

 errors. 



An encoder first encodes the message according to (32), where dj 

 represents the jth digit of the characteristic of Xi , 



J2 CijXi = mod h. (32) 



The decoder calculates the corrector using the following formula where 

 Xi represents the received value of Xi ; 



XI Cij^i = Cj mod h. (33) 



The decoder then examines the digits of the corrector in order. The 

 first digit which is not shows the magnitude, d, of the error. All digits 

 are then divided by d (provided d 7^ 0). (That division is unique, as 

 shown by (30).) The result of this division is the characteristic of the 

 incorrect digit, which is then corrected by subtracting d. 



Consider a code for correcting a single unrestricted error in a six digit 

 message for a base 5 channel: 



6 ^ ''—^. (34) 



A value of 2 for m will satisfy equation (34). The characteristics are 14, 

 13, 12, 11, 10 and 01, the last two being check digit characteristics, for 

 Xi , X2 , X3 , Xi , X5 , and X6 respectively. Here, Xi , X2 , Xs , and Xi are in- 

 formation digits. The encoding formulas are: 



xi + .T2 + X3 + .T4 = —Xi mod 5, (35) 



Axi -f 'Sx'i -f 2x3 + Xi = — .Te mod 5. (36) 



The decoding and correcting formulas are: {x/ is the received value of 



Xi) 



Xi + x-/ + X3 + Xi + Xr/ = ci mod 5, (37) 



4.t/ -\- 3x2' + 2x/ + Xi' + .Te' = C2 mod 5. (38) 



The corrector is C1C2 . 



