ERROR DETECTING AND CORRECTING CODES 



153 



Table III 



As an illustration of how this code "works" let us take the symbol 

 11110 corresponding to the decimal value 12 and change the 1 in 

 the fifth position to a 0. We now examine the new symbol 



1110 



by the methods of this section to see how the error is located. From Table II 

 the first parity check is over positions 1, 3, 5, 7 and predicts a 1 for the first 

 position while we find a there ; hence we write a 



1 . 



The second parity check is over positions 2, 3, 6, 7, and predicts the second 

 position correctly; hence we write a to the left of the 1, obtaining 



1 . 



The third parity check is over positions 4, 5, 6, 7 and predicts wrongly; hence 

 we write a 1 to the left of the 1, obtaining 



10 1. 



This sequence of O's and I's regarded as a binary number is the number 5; 

 hence the error is in the fifth position. The correct symbol is therefore ob- 

 tained by changing the in the fifth position to a 1. 



4. Single Error Correcting Plus Double Error Detecting Codes 



To construct a single error correcting plus double error detecting code we 

 begin with a single error correcting code. To this code we add one more posi- 



