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We present our preliminary results of the non-perturbative determination ofthe valence mass dependent coefficients $b_\mathrm{A}-b_\mathrm{P}$ and$b_\mathrm{m}$ as well as the ratio $Z_\mathrm{P} Z_\mathrm{m}/ Z_\mathrm{A}$entering the flavour non-singlet PCAC relation in lattice QCD with $N_f=3$dynamical flavours. We apply the method proposed in the past for quenchedapproximation and $N_f=2$ cases, employing a set of finite-volume ALPHAconfigurations with Schr\"odinger functional boundary conditions, generatedwith $O(a)$ improved Wilson fermions and the tree-level Symanzik-improved gaugeaction for a range of couplings relevant for simulations at lattice spacings ofabout $0.09 \,$fm and below.

Non-perturbative determination of improvement $b$-coefficients in $N_f=3$