13()0 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



Table V — Decimal Error Correction Codes 



* The single digit me.s.sage containing the points 0, 3, 6, 9 is an exception to 

 the inequality a/3 ^ h, because the mixed digit checks only itself. 



For the most efficient code b"7Q should be minimized. This term repre- 

 sents the ratio of the number of possible messages for an n digit message 

 with and without error correction. This is normally at least as great as 

 2n + 1, the number of possible corrections on such a message. 



Table V shows the most efficient decimal codes of this type for an n 

 digit message, for values of n from 1 to 20. Where two or more different 

 codes are eciually efficient, the code with the fewest mixed digits is shown. 

 It is easy to convert from a code using two mixed digits with ai = 5, 

 a-i = 2, to one using a check digit with a = 10, or to make the inverse 

 conversion, and to show that both codes are equally efficient. 



IV. SINGLE ERROR CORRECTION CODES, UNRESTRICTED ERROR 



The problem of correcting an unrestricted error on one digit of a 

 message must be divided into two categories, depending on whether h 

 is a prime number or a composite number. As will be seen, the error 

 correction problem for prime bases is considerably simpler than that for 

 composite bases. The method for correcting errors in prime number 

 systems was discovered by Golay,^ although this did not come to the 

 author's attention until after he had worked out the same method. The 



