528 THE BELL SYSTEM TECHXICAL JOURXAL, MARCH 1957 



where we define Vs = for s < and where the coefficients kp,s are 



(The derivation of (36) is given in Appendix A.) Equations (36), written 

 out, are of the form 



'i"2, e-lJ'e-l 'i'2. « »'e + ^"2, e+ll'e+l — ""2, e+2»'e+2 — 



from which we see that (36) may be used to determine 



Ve+l , Ve+2 , ' " • 



recursively in terms of 

 We see also that if 



Ve = Ve-l = • • • = Py^i = 



(with 7 such that O^T^e — 1) then 



Ve+l = Ve+2 = ■ • • = V-ie-y = 0. 



This has the following interpretation in terms of e-codes. It is well known 

 (and obvious) that two different code words in an e-code must be sepa- 

 rated by distance at least 2e + 1. Hence if the code word of least weight 

 in an e-code is of weight 7 then all other code words are of weight not 

 less than 2e -f 1 — 7. In the generating function for such a code it must 

 be the case that not only 



but in fact 



G'-'^x) = x-" + 0(.T-^+'-^) (38) 



Equations (36) insure that this condition is satisfied automatically.* 

 As a particular case of (38), we have 



&'\x) = 1 4- Oix''-"'). 



We see that if we apply the operator Ly to G^'^\x) there will result 



LyG''\x) = ^yin, 7i)x' + Oix''^'-') (39) 



* It is also necessary for the existence of an e-code that (36) determine ve+i , 

 ve+2 , • • • as non-negative integers when i>e , Ve~i , • • • , yo are those of (31). This 

 condition is discussed a little further in Appendix A. 



