222 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Pr{Y -. Ai I X -^ Ad = Z Pr{N -^ AiA.Sc) = QiA^A,) 



where Q{Ai) is the sum of the prol)abiUties of the elements that are in 

 the same column as Ai in the standard array. Therefore 



Pr{Y -> .4,) = E Pr{Y -> A, \ X -^ AdPr{X -^ A^ = ^ E QU,A,) 



= 4, since E Q^A.A^ = E QUi) = 1. 

 This last follows from the group property of the alphabet. Therefore 



/i(lO = -- E P>iy -^ A,) log Pr{Y -^ A,) = - bits/symbol. 

 n n 



It follows then from (10) that 



h{X I Y) = h(Y I X) 



The computation of h(Y \ X) follows readily from its definition 



h{Y I X) = E Prix -^ AdhiY \ X -^ Ai) 



i 



= -E Prix -> AdPriY -^ Aj \ X -> Ai) 



log PHY -^Aj I X-^Ai) 

 = -^,1211 PriN ->AiScAj) log E PriN -> AiS„,Aj) 



I 



= -^,ZQiAiAj)'}ogQiAiAj) 



Zi ij 



= - EQU,)logQ(A,) 



i 



Each letter is n binary places. Proposition 2, then follows. 



2.3 DISTANCE AND THE PROOF OF THEOREM 1 



Let A and B be two elements of Bn ■ We define the distance, diA, B), 

 between A and B to be the weight of their product, 



d{A, B) = w(AB) (11) 



The distance between .4 and B is the number of places in which A and 

 B difTer and is jnsl the "Hamming distance." ^ In terms of the n-cube, 

 diA, B) is Ihe minimum mmiber of edges that must be traversed to go 



