BINARY BLOCK CODING 519 



II. DISTRIBUTION OF CODE WORDS 



Suppose an c-code on n places is given. Let us inquire as to the dis- 

 tribution of weights of code words. We denote by v^ the number of code 

 words of weight s, ^ s ^ r?, and by 



G{x) = Z Psx' (2) 



the generating function for these numbers, with x a complex but other- 

 wise free variable. We show in this section that G(x) satisfies a certain 

 inhomogeneous linear differential equation of order e. 



If there exists an e-code on n places then this differential equation will 

 have G(x) as a polynomial solution; the necessary condition for the 

 existence of an e-code on n places given in Section 4 is essentially a 

 restatement of this fact.* First, however, we must derive the differential 

 ecjuation and obtain its solutions. 



If Wa is a code word of the given e-code (1 ^ a ^ K), define the set 

 oi j -neighbors of Wa as the set of n-words which lie at distance exactly 7 

 from iVa ; designate this set by Sj(Wa). (So(Wa) is the set whose only 

 element is Wa itself.) Our derivation is based on the observation that, in 

 an e-code on ?i places, 



U U Sjiw„) = Bn t (3) 



a=l i=0 



is a partition of 5„ . For, the detection regions: 



e 



U SiiWa), I Sa^ K 



3=0 



are disjoint, and in each such sum representing a detection region the 

 summands are disjoint (the distance function being single valued). 

 Furthermore, each n-word of S„ lies in some detection region (close 

 packed property) and hence appears in one of the sets Sj{Wa) for some 

 a and for some j satisfying ^ j ^ e. 

 The set 



U Sj(iv„) 



a = l 



* The author is not yet able to demonstrate the converse. That is, suppose one 

 obtains a polj-nomial sohitionGfi) of (11), below, satisfying; appropriate boundary 

 conditions, and from it some coefficients p^ , ^ s ^ n. It does not follou- from 

 the methods of this article that there is actually some e-code on n places for 

 which these vs represent the number of code words of weight s. 



t U = set union. 



