522 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



1/V^2 about the origin. Tlm.s it follows that G, 3 consists of the ver- 

 tices of an octohedron. 



Error correcting alphabets {{k, K, I))): The error correcting alphabets 

 discussed in Part I can be converted into good alphabets for this channel 

 by replacing all digits which equalled by —1. Three error correcting 

 alphabets appear on the chart; each is labelled by three numbers signi- 

 tying (/v, K, D). 



Slepian alphabets (SD): Using group theoretic methods, D. Slepian 

 has attempted to construct families of alphabets which signal at rates 

 approaching C. Although this goal has not yet been reached, families 

 of alphabets depending on the parameter D have been found which 

 approach the ideal curve to within G.2 db and then get worse as D -^ co. 

 In the simplest of these families of alphabets, D = 2m is even and the 

 letters consist of all the 2'"C2m, m sequences containing m zeros, the 

 remaining places being filled by ztl. The best alphabet in this family 

 is the one with D = 24. It lies 6.23 db away from the ideal curve and 

 contains 1.1 x 10^^ letters. The alphabets of this family for D = 10, 24, 

 and 70 appear on the efficiency graph labelled »S10, /S24, and *S70. 



The conclusion to which one is forced as a result of this investigation is 

 that one cannot signal over a channel with signal to noise level much less 

 than 7 db above the ideal level of Equation (2) without using an un- 

 believably complicated alphabet. No ten digit alphabet tolerates less 

 than 7.7 db more than the ideal signal to noise ratio. 



It would be interesting to know more about good higher dimensional 

 alphabets. They are very much more difficult to obtain. The regular 

 solids, which provided some good alphabets in 3 and 4 dimensions, pro- 

 vide nothing new in 5 or more dimensions ; there are only three of them 

 and they correspond to our D -f I, D; 2D, D, and 2° binary alphabets. 

 Worse still, the packing problem also becomes unmanageable after 

 dimension 5. 



ACKNOWLEDGMENT 



The author wishes to thank R. W. Hamming, L. A. MacColl, B. 

 McMillan, C. E. Shannon, and D. Slepian for many helpful suggestions 

 during the investigation summarized by this paper. 



