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BELL SYSTEM TECHNICAL JOURNAL 



the third parity check 



It remains to decide for each parity check which positions are to contain 

 information and which the check. The choice of the positions 1, 2, 4, 8, • • • 

 for check positions, as given in the following table, has the advantage of 

 making the setting of the check positions independent of each other. All 

 other positions are information positions. Thus we obtain Table II. 



Table II 



As an illustration of the above theory we apply it to the case of a seven- 

 position code. From Table I we find for n = 7, w = 4 and k = ?>. From 

 Table II we find that the first parity check involves positions 1, 3, 5, 7 and 

 is used to determine the value in the first position; the second parity check, 

 positions 2, 3, 6, 7, and determines the value in the second position; and 

 the third parity check, positions 4, 5, 6, 7, and determines the value in posi- 

 tion four. This leaves positions 3, 5, 6, 7 as information positions. The results 

 of writing down all possible binary numbers using positions 3, 5, 6, 7, and 

 then calculating the values in the check positions 1, 2, 4, are shown 

 in Table III. 



Thus a seven-position single error correcting code admits of 16 code sym- 

 bols. There are, of course, 1' — 16 = 112 meaningless symbols. In some ap- 

 plications it may be desirable to drop the first symbol from the code to 

 avoifl the all zero combination as either a code symbol or a code symbol plus 

 a single error, since this might be confused with no message. This would still 

 leave 15 useful code symbols. 



