1370 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



Note that (46), (47) and (48) must be applied consecutively, in that 

 order, since (47) cannot be applied without knowing Xi obtained from 

 (46), and (48) requires Z\ , obtained from (47). 



The characteristics are 01, 11, 21, 31, 41, 51, 61, 71, 81, 91 respec- 

 tively; the complements of the characteristics are 03, 93, 83, 73, 63, 53, 

 43, 33, 23, 13 respectively. The corrector, C1C2 , is calculated at the 

 decoder by the following formulas (x/ is the received value of Xi): 



10 

 ci = E (x^') a - 1) mod 10 (49) 



10 



C2 = S ^i mod 4 (50) 



Consider the example of a message with decimal information digits 

 37520652 and binary information digit 1. Then X'i = 3, Zi = 3, 

 and 7ji = 1, yielding a value of 7 for .I'l . The message is sent as 7 3 3 7 

 5 2 6 5 2. Suppose that the sixth digit is changed to 1 in transmis- 

 sion. Then the corrector has a value 53; this is the complement of the 

 characteristic of the sixth digit and indicates that the sixth digit should 

 be incremented by 1 according to the rules previously stated. If the 

 sixth digit had been received as 1 and the seventh digit also received as 1 

 (an error of +1), then the corrector value would be 10, indicating a 

 double error (see rules stated above) . 



If a multiple of 4 is used as am , the last digit of a characteristic may 

 assume all odd values below am/2. The rule then is that an even value of 

 the last digit of the corrector, or a for the last digit and other digits 

 of the corrector not all 0, indicates a double error. 



The following set of rules and conventions may be used with any 

 base & ^ 4, and anj^ length of message, for deriving a set of charac- 

 teristics for a semi-systematic code for correcting single small errors and 

 detecting double small errors. Since the conventions restrict the effi- 

 ciency of the code, it is conceivable that a different set of conventions 

 will yield a more efficient code in some cases; (51) may be modified 

 through the use of an alternate set of conventions. 



Rule 1. No two digits may have identical characteristics. 



Convention 1 . Choose for a^ a multiple of 4. Let am/4 = g. 



Convention 2. The characteristic of the mixed digit associated with 

 am contains a single 1 in the last position; the rest of its digits are 0. 



Convention 3. The characteristics of the jih mixed or check digit con- 

 tains a 1 in the last position, a 1 in the jth position and O's elsewhere. 



Convention 4- The characteristic of an information digit has an odd 



