There is no strongly regular graph with parameters (99,42,21,15)

We prove, with computer assistance, that no strongly regular graph with parameters (99,42,21,15) exists. At the time of writing, this is one of the parameter sets with at most 100 vertices listed as open in Brouwer's tables. The proof assumes no symmetry. It follows the lattice method of the companion paper on srg(69,20,7,5). • The Gram matrix 3I + A + J has rank 22. The vertices span an even lattice L. The vector y = s/36, where s is the sum of the vertex vectors, lies in the dual of every maximal even overlattice N ⊇ L but not in N.• For x ∈ N^# the pairings ⟨x, v_i⟩ with the vertices satisfy moment and neighbourhood identities. From these we prove by hand that 16 small norms (among others) cannot occur in N^#, and we restrict the values of ⟨y, x⟩ for a few more.• The discriminant form of N is one of four explicit forms. Gluing N + ℤy with a lattice of rank 1, 2 or 3 gives an odd unimodular lattice of dimension 23, 24 or 25, which we take from Borcherds' classification. Three of the four cases are impossible. In the fourth, the host is one of two explicit lattices.• In each of the two hosts, an exhaustive enumeration of the candidates for y (the vectors of norm 11, about 4.3·10^8 per host up to sign), certified by exact vector counts, finds none that satisfies the restrictions. Every restriction used in the final certificates is proved in the text. Every computation on which the proof depends was carried out by at least two independent implementations; the final enumeration was carried out by three. The exception is the construction of lattices from Borcherds' data, which is checked against his root systems, heights and the mass formula. We list the external inputs and the remaining weaker points in Section 13.