156 BELL SYSTEM TECHNICAL JOURNAL 



the same as that of finding subsets of points in the space which maintain at 

 least the minimum distance condition. The special codes in sections 2, 3, 

 and 4 were merely descriptions of how to choose a particular subset of points 

 for minimum distances 2, 3, and 4 respectively. 



It should perhaps be noted that, at a given minimum distance, some of 

 the correctability may be exchanged for more detectability. For example, a 

 subset with minimum distance 5 may be used for: 



a. double error correction, (with, of course, double error detection). 



b. single error correction plus triple error detection. 



c. quadruple error detection. 



Returning for the moment to the particular codes constructed in Part I 

 we note that any interchanges of positions in a code do not change the code 

 in any essential way. Neither does interchanging the O's and I's in any posi- 

 tion, a process usually called complementing. This idea is made more precise 

 in the following definition: 



Definition. Two codes are said to be equivalent to each other if, by a finite 

 number of the following operations, one can be transformed into the other: 



1. The interchange of any two positions in the code symbols, 



2, The complementing of the values in any position in the code symbols. 

 This is a formal equivalence relation (~') since A '^ A\ A ^^ B implies 

 B '^ A\ and A ^^ B, B '~^ C implies A ^^ C. Thus we can reduce the study 

 of a class of codes to the study of typical members of each equivalence class. 



In terms of the geometric model, equivalence transformations amount to 

 rotations and reflections of the unit cube. 



6. Single Error Detecting Codes 



The problem studied in this section is that of packing the maximum num- 

 ber of points in a unit w-dimensional cube such that no two points are closer 

 than 2 units from each other. We shall show that, as in section 2, 2"~ points 

 can be so packed, and, further, that any such optimal packing is equivalent 

 to that used in section 2. 



To prove these statements we first observe that the vertices of the n- 

 dimensional cube are composed of those of two {n — l)-dimensional cubes. 

 Let A be the maximum number of points packed in the original cube. Then 

 one of the two (w — l)-dimensional cubes has at least A/2 points. This cube 

 being again decomposed into two lower dimensional cubes, we find that one 

 of them has at least A/2^ points. Continuing in this way we come to a two- 

 dimensional cube having A/2''~ points. We now observe that a square can 

 have at most two points separated by at least two units; hence the original 

 w-dimensional cube had at most 2"~^ points not less than two units apart. 



