BINARY BLOCK CODING 527 



Since G'^^\x) must be a polynomial in x, it must be expressible as a 

 polynomial in 1 -i- x; hence each root ^^3 must be an integer.* (It is not 

 necessary to require further that ^ ^^ ^ n, since it follows easily from 

 (14) that any real root of <pe(n — 1, ^) satisfies ^ ^i3 ^ n — 1 provided 

 n and e are integers such that ^ e S n.) 



As a corollary we have that if c is odd then n must be odd. This follows 

 from the theorem and the fact that f (n — 1) is a root of (pe(7i — 1, ^) 

 when e is odd, from (17). 



We consider next the case 7 = 0. If 00- • -0 is a code word, then its 

 ^-neighbors are the n-words of weight e. Furthermore, the n-words of 

 weight less than e belong to the code word 00- • -0, and can be e-neigh- 

 bors neither of 00- • -0 nor of any other code word. Hence it must be 

 true that 



G^ix) = \^^ j x" + Oix'-"') (33) 



V 



With Ge^°\x) represented in the form (27), divide the factor ^o(n, ^)d^°\^) 

 in the numerator by the denominator; the result will be 



<Pe(n, ^)d''\^) = [(t - nWn - 1, ^MO + r(0 (34) 



with quotient q(^) a polynomial of degree e — 1 and remainder r(^) a 

 polynomial of formal degree e. The term involving q(^) obviously con- 

 tributes nothing to Ge^'^^x) in (27), so that from (33) and arguments 

 similar to those giving G^^\x), above, r(^) must be the constant 



<P. 



in, n) 



From (34) we then obtain the values of d^^\^) at the poles of the inte- 

 grand in (24), and thus 



g(°)(^) - (1 + ^y |, V ^eCn, n)(l + xYM - xy-'^ 

 <Pe{n - \,n) 0=1 ipe{n,^p){^0 - n)^e'{n - 1,^^) 



Before obtaining G'^''\x) explicitly for intermediate values of 7, we 

 must first discuss a certain set of recursion relations holding between the 

 coefficients Vs of any solution of (11). These relations are 



E (-lyX.Vs =0, p = 1,2, ..., (36) 



s=e— p+l 



* The condition of Golay for the existence of group codes, obtained bv different 

 means, is essentially that <p«(/i — 1, have at least one root an integer. C'f.: (4) 

 of Reference 4, in view of (16), above. 



