BINARY BLOCK CODING 525 



and that 



fj.s = 5—. / 77 TX—fZ rT\ ^^' s = 0, 1, • • • (28) 



(An interesting reciprocity Vj,s = Vs,j is apparent from (28). In an e-code 

 one has (number of j-neighbors of weight s) = (number of s-neighbors 

 of weight j) only for ^ s, j ^ e, since LjG(x) is the generating func- 

 tion for j-neighbors onl}^ if ^ ^ ^ e.) 



IV. BOUNDARY CONDITIONS 



The coefficients Vs , (25), of any solution of (11) satisfy the relation: 



vo 



+ ,,+ ...+,,= ![ ^^d^ = 1 (29) 



Ziri Jt t — n 



by virtue of (16) and the normalizing condition 6(n) = 1. 

 With 7 an integer such that ^ 7 ^ e, denote by 



G''\x) = J: Ps'^'x' (30) 



s=0 



a solution of (11) which satisfies the e boundary conditions 

 (t) (7) (7) n 



(7) (7) (7) n 



(31) 



1' " = f Y+2' "=•••= »'e' " =0 



We must have Vj''^ = 1 in such a solution, from (29). Thus the condi- 

 tions (31) are equivalent to specifying the values of G^^ (x) and its first 

 e — 1 derivatives at the ordinaiy point x = of (11), so that such a 

 solution G''''^{x) exists and is uniquely determined.^ 



Given an e-code on n places, each n-word of B„ lies at distance e or 

 less from exactly one code word; namely, the code word to which it 

 belongs. In particular, the ?i-word 00- • -0 must lie at distance e or less 

 from a single code word. That is to say, there is exactl}'^ one code word 

 in the sphere of radius e centered at 00- • -0. If this code word is of 

 weight 7, then the generating function for the given e-code can be none 

 other than the solution G'''^\x) of (11) defined in the preceding para- 

 graph. 



If there exists an e-code on n places in which the code word of least 

 weight is of weight 7, then there can l)e derived from it an e-code on n 

 places in which the code word of least weight is of weight 7', where 7' 

 is any integer satisfying ^ 7' ^ e. The transformation is that of choos- 

 ing certain places among n and then changing the letters of each //-word 

 of B„ in these places, O's to I's and I's to O's. (Such a transformation 



