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Following Craven and Rouquier's computational method to tackle Brou\'e'sabelian defect group conjecture, we present two algorithms implementing thatprocedure in the case of principal blocks of defect $D \cong C_{\ell} \timesC_{\ell}$ for a prime $\ell$; the first describes a stable equivalence between$B_0(G)$ and $B_0(N_G(D))$, and the second tries to lift a such stableequivalence to a perverse derived equivalence. As an application, we show thatBrou\'e's conjecture holds for the principal $5$-block of the simple group$\Omega^{+}_8(2)$.

An algorithmic approach to perverse derived equivalences: Brou\'e's Conjecture for $\Omega^{+}_8(2)$