Combinatorial Mathematics V: Proceedings of the Fifth by Wilfried Imrich (auth.), Charles H. C. Little (eds.)

By Wilfried Imrich (auth.), Charles H. C. Little (eds.)

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Additional info for Combinatorial Mathematics V: Proceedings of the Fifth Australian Conference, Held at the Royal Melbourne Institute of Technology, August 24 – 26, 1976

Example text

31 Now let ak, m denote the number of different labeled acyclic digraphs on k+m points, exactly k of them out-polnts. The number of ways of extending with k new out-points a labeled acyclic digraph on m points of which exactly s are out-points is k+m (3) [ k )(2 by the reasoning above. ,s^k(m-s) -~j ~ a S ~m--s Summing over s gives ak, m in the form ~, k+m 2k_l)S2k(m-s) as,m-s' ak'm : O0, this s,O yields a straightforward method of computing ak, m in terms of the numbers as, n for for p > O One can find a With the initial conditions a by the sum P aP (5) = 0

That fl = f2 = f3 " Computer Results. It is not easy to combine theoretical with those on bit patterns. attractiveness. Jauregui used 75 hours of CPU time to search actually obtained the result for n = 26 by Andres uses i minute of CPU time for Briefly, results on the frequency of A computer approach has, for small by hand). n = 26 (Golay had n = 26, and clearly can go farther. and X2 = ( E G H F ) be 2 quads, with a quad product defined as XIX 2 = I(AE + BF + CG + DH). l's a certain The algorithm employed the idea is to let XI = ( ~ ) n, 51 Then the total a u t o c o r r e l a t i o n can be w r i t t e n as ~xx(k) = z x i xi+ k = 2L ~k0 Basically, w e use the quads ,R= ,S= ii ,T= -i and their negatives Q, R, , -i S, T.

If we call these can be obtained M(i,j) and N(i,j), respectively, (often not uniquely). B = 0, M(i,j) and N(i,j) The most useful result is that for -i j = 2. that is, the result that (note that this immediately since each quad contributes implies zero to the dot product). Usually, sequences (make A results on bit patterns circular by having A1 are given in terms of cyclic Golay follow An )" With this convention, we quote a sample result. For a pair of cyclic Golay sequences, length 1 is exactly the total number of blocks of L.

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