Cyclic Difference Sets by Leonard D. Baumert

By Leonard D. Baumert

Publication by means of Baumert, Leonard D.

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X ~)(x ~) o. (x -

30)]. Let ~ 2B(i)x i - \ 2 a 2B(O) - 2a - 2 B ( j q # - i r for j = l,j' q TM) + j : i, (j' for i yields ~* + qry 2 la-bl i. modulo ~-i m-i r and r [This of xw - 1 = ~r, and let C(i)x i s = 2, d = i, - 2)q~-lr TM ~ b - 2 B ( q # r m-1 + j q # - i r m ) ]a I _< 2v/w. of q) Further, _( 2v/w. q otherwise. 15, that equation and 8(x) ~ Zi= lw-1 B(i) x i &(wq -1) •(wr -1) C(i) = 0 for q + 2b x = ~. and combining these equations From which, ~(i) -- j=l has a zero at thus and of the analoguous i=0 then 2wi/qr ~ = (-1 + -~qr)/2 is a quadratic residue of both of both is a consequence given by equation ~qr = e where is the Jacobi symbol (see Nagell, (i,qr) > 0, = I representation a, b yields _+ 2b @(q + j r ) .

L then a 0 -= a I ~ -.. 37) holds for 0 < j < w - I, integers of m. , that are rational 38 1 1 1 ao l . . 38) I 1 <_l where b. to be i>j . Since F(x) (for x = i) x w-I . + i>j . x +. 1 . (x ~)(x ~) o. (x -

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