160 BELL SYSTEM TECHNICAL JOURNAL 



We have dealt mainly with systematic codes. The existence of non-sys- 

 tematic codes is proved by the following example of a single error correcting 

 code with n — 6. 





 10 10 1 

 10 110 

 1110 

 10 11 



111111. 



The all symbol indicates that any parity check must be an even one. 

 The all 1 symbol indicates that each parity check must involve an even num- 

 ber of positions. A direct comparison indicates that since no two columns 

 are the same the even parity checks must involve four or six positions. An 

 examination of the second symbol, which has three I's in it, indicates that 

 no six-position parity check can exist. Trying now the four-position parity 

 checks we find that 



12 5 6 



2 3 4 5 



are two independent parity checks and that no third one is independent of 

 these two. Two parity checks can at most locate four positions, and, since 

 there are six positions in the code, these two parity checks are not enough 

 to locate any single error. The code is, however, single error correcting since 

 it satisfies the minimum distance condition of three units. 



The only previous work in the field of error correction that has appeared 

 in print, so far as the author is aware, is that of M. J. E. Golay.'* 



* M. J. E. Golay, Correspondence, Notes on Digital Coding, Proceedings of the LR.E., 

 Vol. 37, p. 657, June 1949. 



