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The AMS-02 and HELIX experiments should soon provide $\mathrm{^{10}Be/^9Be}$ cosmic-ray data of unprecedented precision. We propose an analytical formula to quickly and accurately determine $L$ from these data. Our formula is validated against the full calculation performed with the propagation code \usine{}. We compare the constraints on $L$ set by Be/B and $\mathrm{^{10}Be/^9Be}$, relying on updated sets of production cross-sections. The best-fit $L$ from AMS-02 Be/B data is shifted from 5 kpc to 3.8 kpc when using the updated cross-sections. We obtained consistent results from the Be/B analysis with USINE, $L=3.8^{+2.8}_{-1.6}$ kpc (data and cross-section uncertainties), and from the analysis of $\mathrm{^{10}Be/^9Be}$ data with the simplified formula, $L=4.7\pm0.6$ (data uncertainties) $\pm2$ (cross-section uncertainties) kpc. The analytical formula indicates that improvements on $L$ thanks to future data will be limited by production cross-section uncertainties, unless either $\mathrm{^{10}Be/^9Be}$ measurements are extended up to several tens of GeV/n, or nuclear data for the production of $\mathrm{^{10}Be}$ and $\mathrm{^9Be}$ are improved; new data for the production cross-section of $\mathrm{^{16}O}$ into Be isotopes above a few GeV/n are especially desired., 9 pages, 2 tables, 5 figures (matches accepted A&A version): comparison to [De21] moved to appendix (also highlight impact of reacceleration)

On-going measurements of the cosmic radiation (nuclear, electronic, and gamma-ray) are shedding new light on cosmic-ray physics. A comprehensive picture of these data relies on an accurate determination of the transport and source parameters of propagation models. A Markov Chain Monte Carlo is used to obtain these parameters in a diffusion model. From the measurement of the B/C ratio and radioactive cosmic-ray clocks, we calculate their probability density functions, with a special emphasis on the halo size L of the Galaxy and the local underdense bubble of size r_h. The analysis relies on the USINE code for propagation and on a Markov Chain Monte Carlo technique (Putze et al. 2009, paper I of this series) for the parameter determination. As found in previous studies, the B/C best-fit model favours diffusion/convection/reacceleration (Model III) over diffusion/reacceleration (Model II). A combined fit on B/C and the isotopic ratios (10Be/9Be, 26Al/27Al, 36Cl/Cl) leads to L ~ 8 kpc and r_h ~ 120 pc for the best-fit Model III. This value for r_h is consistent with direct measurements of the local interstallar medium. For Model II, L ~ 4 kpc and r_h is consistent with zero. We showed the potential and usefulness of the Markov Chain Monte Carlo technique in the analysis of cosmic-ray measurements in diffusion models. The size of the diffusive halo depends crucially on the value of the diffusion slope delta, and also on the presence/absence of the local underdensity damping effect on radioactive nuclei. More precise data from on-going experiments are expected to clarify this issue., Comment: 20 pages, 14 figures, minor language corrections to match the A&A accepted version