ERROR DETECTING AND CORRECTING CODES 159 



drop one coordinate from the n dimensional code. This reduces the minimum 

 distance of 2k to 2^ — 1 while leaving N the same. It is clear that if one 

 code is of minimum redundancy then the other is, too. 



9. Miscellaneous Observations 



For the next case, minimum distance of five units, one can surround each 

 code point by a sphere of radius 2. Each sphere will contain 



1 + du, 1) + C(n, 2) 



points, where C{n, k) is the binomial coefficient, so that an upper bound on 

 the number of code points in a systematic code is 



2" 2"+^ 



1 + C{n, 1) + C{n, 2) n^ + « + 2 



> T 



This bound is too high. For example, in the case of n — 7, we find that 

 w = 2 so that there should be a code with four code points. The maximum 

 possible, as can be easily found by trial and error, is two. 



In a similar fashion a bound on the number of code points may be found 

 whenever the minimum distance between code points is an odd number. 

 A bound on the even cases can then be found by use of the general theorem 

 of the preceding section. These bounds are, in general, too high, as the above 

 example shows. 



If we write the bound on the number of code points in a unit cube of dimen- 

 sion n and with minimum distance d between them as B{ii, d), then the 

 information of this type in the present paper may be summarized as follows: 



B{n, 1) = l"" 



Bin, 2) = r-' 



Bin, 3) = 2™ < 



2" 



Bin, 4) = 2" < 



I, 



Bin - 1, 2/^ - 1) = Bin, 2k) 

 Bin, 2/^ - 1) = 2" < 



n + 1 



2«-i 



1 + cin, 1) + . . . + c(«, k - i; 



While these bounds have been attained for certain cases, no general 

 methods have yet been found for contructing optimal codes when the mini- 

 mum distance between code points exceeds four units, nor is it known 

 whether the bound is or is not attainable by systematic codes. 



