BINAEY BLOCK CODING 529 



using the differential operator form for Ly, (9). On the other hand, the 

 function 



iPy{n, n) 2x1 Jt (^ - n)ipe(n - 1, ^) L <Py{n,n) J 



is a solution of differential equation (11), in view of the discussion fol- 

 lowing (24). From (39) we see that this function can be none other 

 than G'''^\x). Finall}^, applying Ly to G^'^\x) in the form (35), we have 

 explicitly 



r(7)/.A __ (1 + a:)" , <pe{n, n) 



G'->\x) = V ' 7 , + 



(Pe(w - l,n) <py(n,n) /?=i 



(41) 

 <Py(n, ^,)(1 + x)^^a - xy-^^ 0<7<6 



^c{n,^^)i^? - n)iPe'in - 1,|^) 



V. EXAMPLES 



The known cases where the condition of Hamming is satisfied are the 

 f ollo\ving : 



Case I: e = 0, n ^ 1 



The Hamming expression (1) reduces to unity. In fact, 



Mn - 1, ^) = 1, 



and the condition that all roots be integers is vacuous. The generating 

 function for code words is (uniquely) : 



G {X) = — — - = (1 + X) 



(Po{n — 1, 71) 



Each ?i-word of 5„ is a detection region and thus a code word. There is 

 no error correction. 



Case II: e ^ 1, n = e 



The Hamming expression becomes the sum of all the terms in the bi- 

 nomial expansion of (1 + 1)". The "codes" in this class consist of a single 

 code word surrounded bj^ its detection region consisting of the sphere 

 Bn of radius n. No signalling is possible, of course, but our methods still 

 apply. 



From the representation 



, ,. 1 /- (1 + xYa - xy-^ , 



Mn, ^) = ;5— . / ^— ZT dx 



2iri Jc X^+^ 



(42) 



2tI Jc V 



(1 + 2vy 



*+Hi + vy-'+^ 



dv 



