154 BELL SYSTEM TECHNICAL JOURNAL 



tion for checking all the previous positions, using an even parity check. To 

 see the operation of this code we have to examine a number of cases: 



1. No errors. All parity checks, including the last, are satisfied. 



2. Single error. The last parity check fails in all such situations whether 

 the error be in the information, the original check positions, or the last 

 check position. The original checking number gives the position of the 

 error, where now the zero value means the last check position. 



3. Two errors. In all such situations the last parity check is satisfied, and 

 the checking number indicates some kind of error. 



As an illustration let us construct an eight-position code from the previous 

 seven-position code. To do this we add an eighth position which is chosen 

 so that there are an even number of I's in the eight positions. Thus we add 

 an eighth column to Table III which has: 



Table IV 

 

 

 1 

 1 



1 

 1 

 

 



1 

 1 

 

 





 

 1 

 1 



PART II 

 GENERAL THEORY 



5. A Geometrical Model 



When examining various problems connected with error detecting and 

 correcting codes it is often convenient to introduce a geometric model. 

 The model used here consists in identifying the various sequences of O's and 

 I's which are the symbols of a code with vertices of a unit //-dimensional 

 cube. The code points, labelled .v, y, z, ■ ■ • , form a subset of the set of all 

 vertices of the cube. 



Into this space of 2" j^oints we introduce a dislcDicc, or, as it is usually 

 called, a metric, D{x, y). The delinition of the metric is based on the observa- 

 tion that a single error in a code point changes one coordinate, two errors, 

 two coordinates, and in general d errors produce a diflference in d coordinates. 



