BIXARY BLOCK CODIXG 535 



Rearrange the left hand member and re-expand it, to get 



[(1 - x) - yd + x)r = [(1 - y) - x{l + y)r 



= i:(-lr(',')•^•^a + z/r(l - 



^)- 



(B3) 



n n 



71 



s=0 r=0 \S / 



Comparmg coefficients of x^y"^ in (B2) and (B3), we have, finally, 



(-!)'■ (^^) ^.(n, r) = ( - D' (^^ ^Xn, s) (n, r, s integers), (B4) 

 or, changing notation slighth', 



Vi(n - 1, e) = (-1)'- y. ^ ,( v.(n - 1, {) (Bo) 



(" ; ') 



(with n, e, ^ integers and ^ e, ^ ^ n — 1). Thus ii<pe{n — 1, ^) vanishes 

 for e different integers ^ then so must (p^n — 1, c), at least when e < n. 

 But v'j(w — 1, e) is the coefficient of x in (1 + .t)*(1 — .t)"^^~* when this 

 is written out as a polynomial in x, by definition. 



REFERENCES 



1. R. W. Hamming, B.S.T.J., 29, p. 147, 1950. 



2. C. E. Shannon, B.S.T.J., 27, p. 379, 1948. 



3. P. Elias, Trans. I.R.E., PGIT.4, p. 29, 1954. 



4. M. J. E. Golay, Trans. LR.E., PGIT-4, p. 23, 1954. 



5. D. Slepian, B.S.T.J., 35, p. 203, 1956. 



6. E. L. Ince, Ordinary Differential Equations (Dover), Ch. V, XV. 



7. U. J. E. Golav, Proc. I.R.E., 37, p. 637, 1949. 



8. K. M. Case, Phys. Rev., 97, p. 810, 1955. 



