Published online by Cambridge University Press: 22 January 2016
The 2g theta constants of second kind of genus g generate a graded ring of dimension g(g + 1)/2. In the case g ≥ 3 there must exist algebraic relations. In genus g = 3 it is known that there is one defining relation. In this paper we give a relation in the case g = 4. It is of degree 24 and has the remarkable property that it is invariant under the full Siegel modular group and whose Φ-image is not zero. Our relation is obtained as a linear combination of code polynomials of the 9 self-dual doubly-even codes of length 24.