N'OX-BIXARY ERROR CORRECTION CODES 1361 



adaptation to non-prime channel bases is believed to be novel. Since the 

 adaptation makes use of the code for prime bases, both will be described. 



4.1 Prime Number Base, Single Unrestricted Error Correction Code 



This code depends upon a fundamental property of prime numbers, 

 well known in number theory.^ Let p represent a prime number and d, 

 c, and w represent non-negative integers less than p, related by the 

 expression : 



dw = c mod p. (30) 



If d 9^ 0, then d and c uniquely determine w. 



In order to have a simple characteristic systematic code for correcting 

 unrestricted errors, it is necessary and sufficient that the set of charac- 

 teristics shall have the property that all multiples of all characteristics 

 are distinct. Equation (30) implies a unique correspondence between mul- 

 tiples of a characteristic and the characteristic itself, if we consider c to 

 be the multiple, d the multiplying factor and w a digit of the charac- 

 teristic. An error, d, is simply identifiable if a known digit of a charac- 

 teristic is always 1. If each characteristic is distmct from everj' other and 

 if a sufficient number of check digits are available, a simple characteristic 

 code can be obtained. In the following set of rules and con^•entions which 

 may be used for deriving a set of characteristics for a simple charac- 

 teristic systematic code for correcting single unrestricted errors, p repre- 

 sents the prime number base of the channel. The number base of the 

 channel must be prime, and the length of the message is arbitrary. Since 

 the rules and not the conventions limit the efficiency of the code, no other 

 set of conventions may be found which will lead to a more efficient code 

 of this class. 



Rule 1. For an n digit message, m check digits are required and m 

 must satisfj^ the inequalitj- 



n ^ ^-^. (31) 



Rule 2. Each digit must have a difi"erent characteristic. 



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

 order, i.e., CuCa • • • dm • The first digit which is not must be 1. 



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

 jth position and O's elsewhere. 



Rule 1 is required for a code for correcting single unrestricted errors 

 since any digit must be correctable in one of p — 1 ways. This implies 

 a minimum of n(p — 1) + 1 states for the corrector, one for each cor- 



