506 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



into a binary sequence composed of repetitions of the three letters. 



00000 



11100 



and 00111 



If K = 2°, the alphabet consists of all binary sequences of length D 

 and hence if any of the digits of a letter is altered b}^ noise the letter Avill 

 be misinterpreted at the receiving end of the channel. If K is somewhat 

 smaller than 2^ it is possible to choose the letters so that certain kinds 

 of errors introduced by the noise do not cause a misinterpretation at 

 the receiver. For example, in the three letter alphabet given above, if 

 only one of the five digits is incorrect there will be just one letter (the 

 correct one) which agrees with the received secjuence in all but one place. 

 More generally if the letters of the alphabet are selected so that each 

 letter differs from every other in at least 2A- + 1 out of the D places, 

 then when A- or fewer errors are made the correct interpretation of the 

 received sequence will be the (uniciue) letter of the alphabet which 

 differs from the recei\'ed seciuence in no more than A' places. An alphabet 

 with this property will be called a A- error correcting alphabet . 



Error correcting alphabets have the advantage over the random 

 alphabets which Shannon used to prove his encoding theorems that they 

 are uniformly reliable whereas tShannon's alphabets are reliable only in 

 an average sense. That is, Shannon proved that the probability that a 

 letter chosen at random shall be received incorrectly can be made ar- 

 bitrarily small. However, a certain small fraction of the letters of Shan- 

 non's alphabets are allowed a much higher probability of error than the 

 average. This kind of alphabet would be undesii-able in applications such 

 as the signalling of telephone numbers; one would not want to give a 

 few subscribers telephone numbers which are received incorrectly more 

 often than most of the others. It is only conjectured that the rate C can 

 be approached using error correcting alphabets. The alphabets which are 

 to be considered here are all error correcting alphabets. 



A geometric picture of an alphabet is obtained by regarding the D 

 digits of a sequence as coordinates of a point in Euclidean D dimensional 

 space. The possible received sequences are represented by vertices of 

 the unit cube. A A- error correcting alphabet is represented by a set of 

 vertices, such that each pair of vertices is separated by a distance at 

 least \/2k + 1 



Let Ko{D, A) be the largest number of letters which a D dimensional 



^ R. W. Hamming, "Error Detecting and Error Correcting Codes," Bell System 

 Tech. J., 29, pp. U7-160, 1950. 



