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Standing on the inner product relations for the even unimodular lattices, which were founded by Schöneberg and Hecke and were extensively used by B. Venkov, we adapt their formulas so as to compute the Fourier coefficients of Siegel theta series of degrees two, three, four and five associated with the Leech lattice. As by-products some combinatorial structures, such as spherical codes or the association schemes, are deduced in a systematic way.

SIEGEL THETA SERIES OF VARIOUS DEGREES FOR THE LEECH LATTICE