530 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



(valid for all n, ^) we have immediately 



<p,Xn - 1, t) = 2" (^) = ?! ^(^ - 1) . . . (^ - n + 1) 



and the roots are 0, 1, •••,« — 1. The generating function 6'^^\x) of 

 (32) becomes* 



2" ItI Jt (?)n4-l 



= ^ 1(1 + -i-) - (1 - ■■'■)r = x' 



as one might expect. The explicit form for (pn{n, ^) is somewhat com- 

 plicated, but for ^ an integer it follows immediately from definition (13) 

 that 



<Pn{n,k) = (-1)""' ^ = 0, 1, ••• ,n 

 From (35), then, 



= :^ [(1 + x) + (1 - .r)]" = 1 



which, again, is not surprising. The details for other values of y seem to 

 be more tedious, although one expects (41) to yield G^'^\x) = a;\ 



Case III: e ^ 1, n = 2e + 1 



The Hamming expression in this case: 



l+(2« + l)+...+(2«+l).2^. 



consists of the first half of the terms in the binomial expansion of 

 (1 + 1) * \ The code words in a code of this class are any two ?i-words 

 separated by distance n (i.e., two vertices at opposite corners of the 

 n-cube). The group codes in this class are the "majority rule" codes. f 

 From (42) we have (using the substitution y = 4v -f 4v ) 



(f).-.!(0 



(lonotes the descending factorial. 



t The two code words in such a code are 00 • • • and 11 • • • 1. An n-word he- 

 longs to 00 • • • if it contains more O's than I's, and to 11 • • • 1 if it contains more 

 I's than O's. 



