COMPAKISOX OF .SIGNALLING A Ll'llA HKTS 521 



point has 18 nearest neighbors. A hyi)ersphei'o of radius 3 a))Out tlie 

 orijiiii has a volume (7r"/2)3 , ahoul 100. Thus it contains about 800 

 kiltiee points. Take these as the code points of the 800, 4 code. Their 

 average squared distances from the origin can be estimated as 



,.3 

 5 



r dr 



JQ 



^3 



\ r'dr 

 Jo 



= I m' = G. 



A'' in Equation (13) may be estimated at 18; this is conservative because 

 some lattice points outside the spliere arc being counted. 



The two remaining four dimensional alphabets l)elong to two families 

 of Z)-dimensional alphabets. 



The 4, 3; 5, 4; • • • ; Z) + 1, -D • • • alphabets are the vertices of the 

 simplest regular solid in /^-dimensional space. For example, 4, 3 is a 

 tetrahedron. Such a solid can be constructed from D + 1 vertices whose 

 coordinates are the first Z) + 1 rows of the scheme 



••• 



10 ••• 



2V3 

 1 



2\/3 2V6 



1 1 5 



2\/3 2\/6 2vl0 



1 1 1 



2\/3 2Vg 2\/iO 



The vertices all lie a distance \/D/2(D + 1) from the centroid of the 

 figure. 



6, 3; 8, 4; • • • ; 2D, D, • • • are obtained by placing a point wherever 

 any positive or negative coordinate axis intersects the sphere of radius 



