NON-BINARY ERROR CORRECTION CODES 1367 



message had been in a unodecimal code. (This has been illustrated in 

 Section 4.1.) 



The first step of the encoding procedure is to calculate the unadjusted 

 check digits. Next, the adjusted check digits and adjustment digit are 

 selected according to the value of y, the information digit associated with 

 the adjustment. The message is then ready for transmission. 



At the decoder, the message is first corrected as if it had been re- 

 ceived as a unodecimal message. The information digits are then hi 

 their corrected states. Next, the adjustment digit and the check digits 

 are examined and the inverse of the encoding process used to select a 

 particular set of check and adjustment digits is used to reconstruct the 

 value of y which originally controlled the selection. In the example given 

 above, the values of Xw , Xn , Xn are 8, 5, 4 respectively; the decoder recog- 

 nizes that this is the seventh lowest value of Xio , which means that the 

 value of y, used in selecting .Tio and the adjusted values of .ru and Xn , 

 was 6. 



The code described above is fairly efficient; about 90 per cent of the 

 corrector values can be associated with corrections; the product of the 

 information states and the check states is about 97 per cent of the 

 total number of states of a twelve decimal digit word. Each of the above 

 factors reduces the efficiency of the code below a possibly unattainable 

 maximum. It will be noted, however, that this reduction is relatively 

 small in both cases, and is very much lower than would be the case for 

 any of the rejected schemes. The scheme is not difficult to instrument; 

 relatively little additional ec^uipment is required in addition to the 

 basic equipment for instrumenting a simple prime number base chan- 

 nel, unrestricted single error correcting code system. 



The method of adjustment digits is general and can be used for de- 

 riving a single error correction code for correcting unrestricted errors 

 for any channel base. Any convenient prime check base, p, at least as 

 great as h may be used, although the lowest will generally be the most 

 efficient. The only requirements which must be fulfilled are that the 

 number of states of the adjustment digit must be at least 1, and that at 

 least two check digits must be associated with each adjustment digit. 

 An adjustment digit associated with m check digits, working with a 

 channel base 6 and a check base p, may have b — m{p — h) different states. 



V. SINGLE ERROR CORRECTION, DOUBLE ERROR DETECTION CODES FOR 

 CORRECTING SMALL ERRORS 



Single error correction, double error detection codes are ver}^ useful 

 in situations where a message may occasionally be repeated. In order 



