NON-BINARY ERROR CORRECTION CODES 



1369 



detected in an over-all check digit, difficulties would be encountered in 

 determining the direction of the correction, so that the information 

 conveyed by the mixed digit could be used. Actually, means are avail- 

 able, for accomplishing an adaptation of binary techniciues. These meth- 

 ods are described in Section \ll but they are less straightforward than 

 the ones described below. 



For channels with base b, greater than 3, at least one check may be 

 made using a check base, am , that is 4 or greater. If characteristics are 

 used whose last digit (the digit associated with the a,„ check) is always 1, 

 and whose only other limitation is that each characteristic is different 

 from every other characteristic, a satisfactory code is obtained. Single 

 errors are corrected in the normal way. If the last digit of the corrector 

 is 1 or a,n — 1, the error is ±1 respectively on the digit whose charac- 



teristic or whose characteristic complement is indicated by the cor- 

 rector. If the last digit of the corrector is 2 or a^ — 2, or the last digit is 

 and other digits are not all 0, a double error is indicated. If the entire 

 corrector is made up of O's, the message is correct as received. 



An example is a code for a ten digit message, decimal base channel; 

 eight decimal information digits, one mixed digit conveying binary 

 message information (such as the sign of the decimal number) and cjua- 

 ternary (base 4) check information, and one check digit are transmitted 

 in each message. Let Xi and .r^ represent the mixed and check digit re- 

 spectively, Xs through Xio the information digits, yi the binary informa- 

 tion conveyed by Xi , and Zi the quaternary check information conveyed 

 by Xi . The encoding formulas are : 



2.1-3 + 3.^4 + 4i-5 + 5.r6 + 6.^7 -\- 7xs + 8.1-9 + 9.rio = -x-. mod 10, (46) 



x-2 + X3 -f .T4 + .Ts + .Te + .Ty + .^'s + -n + .rio = —zi mod 4, (47) 



xi = z,-{- 42/1 . (48) 



