1358 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



digit in the number base 0j . The maximum vahie of Zj is usually «_, — 1, 

 since Zj is a digit in the number base ay. Thus, 



Xk+i = y,a, + ^, ^ 6 - 1, (24) 



(^, - l)a, + a; - 1 ^ 5 - 1, (25) 



ocj^, ^ h. (26) 



Equation (24) restates (19-j), and also states that the maximum value 

 of any digit x, is 6 — 1, where b is the number base of the channel. In 

 (25), the maximum values of tjj and Zj are substituted to yield the result 

 shown in (26). 



It was stated above that the maximum value of Zj is usually aj — 1. 

 An exception occurs only in case Zj checks only itself and other mixed 

 digits, the latter being restricted to fewer than 6—1 states. Under such 

 circumstances, the value of z is sometimes restricted, so that even though 

 z is calculated to satisfy a check, modulo aj [see equation (19-j)], it can- 

 not assume a> — 1 values. For example, a code for transmitting a 

 single digit message over a decimal channel and permitting the correc- 

 tion of small errors, might use as the set of transmitted messages the 

 digits, 0, 3, 6, 9. In this case, a = 3 (any correct message satisfies the 

 check X = mod 3) and (5 — 4 since four different messages may be 

 transmitted. In this case, z is restricted to the value because the mixed 

 digit checks only itself. 



In order to correct single errors of ±1, using a simple characteristic 

 code, it is necessary and sufficient that every characteristic be different 

 from every other characteristic, and that it also be different from the 

 complement of every other characteristic. 



The following rules and conventions may be used to derive a set of 

 characteristics which meet the requirements for a simple characteristic 

 semi-systematic or systematic code for correcting small errors for any 

 base 6^3 and an arbitrary length message. No set of conventions can 

 be found which will lead to a more efficient code of this class, since the 

 rules, not the conventions limit the efficiency of the code. 



Rule 1 . For an n digit message, including mixed digits, containing ?/? 

 mixed or check digits of which mi are associated with an even modulus, 

 a, the inequality 



(ara-y ... -a,,. - 2"'i)/2 ^ n (27) 



must be satisfied. 



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

 a characteristic different from that associated with any other digit. 



Rule 3. Since the mth check is the last one to be calculated, and the 



