520 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



Fig. 5 are not the code signals themselves but impulse functions which 

 are to be passed through a low pass filter with cutoff at W c.p.s. to form 

 the code signals. 



The best possible higher dimensional alphabets can be described more 

 easily verbally than pictorially. In four dimensions we have found four 

 alphabets. 



The 25,4 alphabet consists of the origin and all 24 points in 4 dimen- 

 sional space having two coordinates equal to zero and the remaining 

 two equal to l/\/2 or — l/'s/2 . Each of the 24 points lies a unit distance 

 away from the origin and its 10 other nearest neighbors; they are, in 

 fact, the vertices of a regular solid. This alphabet has an advantage be- 

 yond its high efficiency. The code signals are composed entirely of posi- 

 tive and negative pulses of fixed energy and so should be easier to 

 generate than most of the other codes which appear in this paper. 



The 800, 4 alphabet is constructed in the following way: Consider a 

 lattice of points throughout the entire 4-dimensional space formed by 

 taking all the linear combinations with integer coefficients of a basic 

 set of four vectors. That is, the lattice points are of the form CiVi + 

 C2V2 + C3V3 + CiVi where Ci , • • • , C4 are integers and the Vi are the four 

 given vectors. In connection with our problem it is of interest to know 

 what lattice, (i.e. what choice of Vi , Vo , Vz , Va) has all lattice points 

 separated at least unit distance from one another and at the same time 

 packs as many points as possible into the space per unit volume. When 

 a solution to this "packing problem" is known, it is clear that a good 

 alphabet can be obtained just by using all the lattice points which are 

 contained inside a hypersphere about the origin as the letter points. 

 Many of the two dimensional alphabets illustrated in the sketches are 

 related in this way to the corresponding tw^o dimensional packing prob- 

 lem (which is solved by letting Vi and v-i be a pair of unit vectors 60° 

 apart) . A solution to the four dimensional packing problem is aff ored by 



''■ = ^' Vr "■ " 



"=^2' °' °' 72 



"*='°' vl' V2- "• 



This lattice contains two points per unit volume (twice as dense as the 

 cubic lattice in which Vi , • • • ,Vi are orthogonal to one another) and each 



