1352 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



In this code all the possible values of C1C2 correspond to the charac- 

 teristic of a digit or the complement of this characteristic, except 00 

 which corresponds to the correct message. (An inspection of equations 

 (4) through (7) reveals that if Xi = Xi for all values of t, the values 

 of Ci and Ci are 0). Thus, we can assign a unique correction to each 

 value of C1C2 . 



The above techniques are extensible to other number bases and dif- 

 ferent length words provided 6, the number base of the channel, is 

 greater than 2. (The equivalent binary channel problem has been treated 

 by Hamming. ) The following set of rules and conventions may be 

 used for deriving a satisfactory set of characteristics for a simple charac- 

 teristic systematic code used to correct single small errors for any length 

 message, and any base, 6^3. The rules must be followed, and the 

 con^'entions (which represent one pair of conventions out of the set of 

 pairs of conventions, which together with Rules 1 and 2 can be used for 

 deriving a code of this class) if followed, will lead to a reasonably simple 

 method for encoding and decoding messages.* Since the rules, not the 

 conventions, limit the efficiency of the code, no set of conventions can 

 be found which will lead to a more efficient code of this class. 



Rule 1. For an n digit message (including check digits), m check dig- 

 its are required and m must satisfy the following ineciualities : 



if 6 is odd, — - — ^ n, (8a) 



if b is even, ^ n. (8b) 



Rule 2. No characteristic may be repeated; i.e., each digit must have 

 a characteristic different from that associated with any other digit. 



Convention 1. The various digits of a characteristic are arranged in a 

 set order; i.e., Cn , Co, , • • • , Cmi • The first digit which is neither zero, 

 nor (in case h is even) 6/2, must be less than 6/2. There must be at least 

 one such digit. 



Convention 2. The characteristic of the 7th check digit has a 1 in the 

 jih. position and O's elsewhere. 



Rule 1 is required since, for a code of this type, we must be prepared to 

 correct any digit in one of two ways (±1). This implies a mininuun of 

 2 n + 1 values of the corrector, one for each possible correction, and one 

 for the case of no corrections. This means that 6"', the number of possible 



I 



* The above distinction between rules and conventions will be observed 

 tliroughoiit tills paper. 



