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In this paper, we introduce the notion of Jacobi polynomials for codes, and establish some fundamental features of it. This notion comes out of considerations on the various invariants of the codes (e.g. [22–24]); it has to do with Jacobi theta-series of positive definite quadratic forms (cf. [9], p. 82). Thus the name ‘Jacobi polynomials for codes’ traces this fact.

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Abstract. 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.