526 THE BELL SYSTEM TECHNICAL JOUUXAL, MARCH 1957 



corresponds to one of the operations of the orthogonal group which 

 leaves invariant the 7i-cube representing 5„ .) Metric properties in B„ 

 are invariant under such a transformation, clearly, and an e-code is 

 transformed into an e-code. Thus if there exists any e-code on n places 

 then (11) must have e + 1 distinct polynomial solutions G^'^^(.r), satis- 

 fying boundary conditions (31) for each case 7 = 0, 1, • • • , e. 



In (25) for the coefficients v^ , move contour T out to a circle sufficiently 

 large that the expansion 



1 _ ^' 1 (const.) 



(I - n)^e{n - l,a 2«|^+i ^^+2 



converges on T. Suppose that the poljaiomial d(^), of formal degree e, is 

 of actual degree /: d{^) = c^^ + 0(|^~'), c 9^ 0, where ^ / ^ e. Then 

 the numerator of the integrand in (25) is of the form: {2^c^''^^/sl) + 

 0(^*^''^~'), and it is clear that 



V, = ^ s ^ e - f - 1 



^-/ = 2f{e- f) ! ^ ^ 



Hence, if ^^^^(^) denotes the polj^nomial which gives G''''\x) in the repre- 

 sentation (24), then d''''\^) must be of actual degree e — 7. 



A particularly simple case is the one 7 = e; the polynomial 0^*^(^) 

 must be of degree zero, and is determined by the normalization as 

 d^'\^) = 1. Thus 



g.-»M=-L./ a + f;-f-; ,, (32) 



2Tri Jr (^ — n)<pe{n — 1, ^) 



From this we have immediately the following 



Theorem: If there exists an e-code on n places then the equation 

 <Pe{n — 1, ^) = in ^ has e distinct integer roots. 



Proof'. If there exists an e-code on n places, then there exists an e-code 

 on n places in which the code word of least weight is of weight e. The 

 solution (32) of (11) must be the generating function for this e-code; 

 hence (32) must reduce to a pol3aiomial of formal degree n. li(pc{n — 1, ^) 

 had multiple roots then noncancelling logarithmic terms (22) would 

 appear in the G'''\x) of (32). Thus <fe(n — 1, ^) must have e distinct 

 roots ^0 , 1 ^ (3 ^ e. Each solution (1 + .r)^^(l - .r)""*" of the homo- 

 geneous equation appears in G''^\x) with non vanishing coefficient: 



1 



As = 



i^e - n)ipe'{n - 1, ^^) 



