A CLASS OF BINARY SIGNALING ALPHABETS 223 



from vertex ^4 to vertex B. The distance so defined is a monotone fnne- 

 tion of the Euchdean distance between vertices. 



It follows from (11) that if C is any element of B„ then 



d{A,B) = cJ(A(\BC) (12) 



This fact shows the detection scheme (8) to be a maximum likelihood 

 detector. By definition of a standard array, one has 



d(Si , I) ^ d(S,Aj , I) for all i and j 



The coset leaders were chosen to make this true. From (12), 



d(S, , I) = d(SiA,„S,- , / .4„.^S,) = d(SiA,n , A,„) 



d(SAj , /) - diS^AjSiAm , I SiAJ = diAjA,n , SiAr.) 



= d{SiAm , A() 



where Af = AjA^ . Substituting these expressions in the inecjuality 

 above yields 



d(SiAm , A„,) ^ d(SiAm , At) for all i, m, I 



This equation says that an arbitrary element in the array (4) is at least 

 as close to the element at the top of its column as it is to any other letter 

 of the alphabet. This is the maximum likelihood property. 



2.4 PROOF OF THEOREM 2 



Again consider an (n, /c) -alphabet as a set of vertices of the unit n-cube. 

 Consider also n mutually perpendicular hyperplanes through the cen- 

 troid of the cube parallel to the coordinate planes. We call these planes 

 "symmetr}^ planes of the cube" and suppose the planes numbered in 

 accordance with the corresponding parallel coordinate planes. 



The reflection of the vertex with coordinates (ai , a^ , • • • , a^ , • • • , a,j) 

 in symmetry plane i yields the vertex of the cube whose coordinates 

 are (ai , oo , ■ • • , a, -j- 1, • • • , 0,0 . More generally, reflecting a given 

 vertex successively in symmetry planes i, j, k, ■ • ■ yields a new vertex 

 whose coordinates differ from the original vertex precisely in places 

 i, j, k ■ ■ ■ . Successive reflections in hyperplanes constitute a transfor- 

 mation that leaves distances between points unaltered and is therefore 

 a "rotation." The rotation obtained by reflecting successively in sym- 

 metry planes ?', j, k, etc. can be represented by an ?i-place symbol having 

 a one in places ?', j, k, etc. and a zero elsewhere. 



We now regard a given {n, /j)-alphabet as generated by operating on 

 the vertex (0, 0, • • ■ , 0) of the cube with a certain collection of 2 ro- 



