534 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



valid for all n, ^, r and all non-negative integers p, t, where 



->'-«=5(:)(:::)C^:;J (^^) 



The coefficients Kp,s(7i, t; i) vanish unless s ^ ^ + p; if n and p — t 

 are non-negative integers then the coefficients Kp,s(n, r; t) are positive 

 integers provided t — p -\- t :^ s and vanish otherwise. In particular, 

 (setting r = 1, t = e), 



2''('' ~ ^)^M - 1,^) = Z {-iy'-%.s<ps{n, 0,P = 1,2,'", (A5) 



\ P / s=e-p+l 



where we define <ps{n, ^) = for s < 0; the kp,, = Kp,s(n, 1; e) are 

 those of (37) of the text. If we multiply v^ of (25) by (-ly^'kp., 

 and sum on s there results (36), since the left hand member of (A5) is 

 a polynomial multiple of the denominator of the integrand in (25). 



If the code word of least weight in an e-code is of weight y, then the 

 first nontrivial one of the (37) is the one for p = e -\- 1 — y, and it 

 gives (since Vy'^'^^ = 1) 



(-J-) _ Ke+l—y,y 



fi^e+l— 7,2e+l— 7 



_ (n — 7) (n — 7 — 1) • • • (n — e) 



~ (2e+ 1 -7)(2e - y) - ■ ■ {e + I) 



A necessary condition for the existence of an e-code on n places is that 

 this expression be a non-negative integer in each case 7 = 0, 1, • • • , e. 

 It is not clear, however, that this condition is independent of the one 

 set forth in the theorem of Section IV. 



Appendix B 



We give here a relation due to K. M. Case* which shows that the 

 statement of our main result as it appears in the Abstract heading this 

 article agrees with the theorem proved in Section IV. 



In the defining relation 



(1 + ,0^(1 _ -,)— = f^ ^.V.(^, r) (Bl) 



s=0 



for the coefficients <Ps{n, r) assume that 7i and r are integers, multiply 

 both sides by ( — l)*" ( ) 7/, and sum on r, ^ r S fi. The result is 



[(1 - x) - yd -f x)T = ti: (-1)^ ('') yx<pXn, r) (B2) 



r=0 s=0 \' / 



