We present a detailed description of the structural characteristics of the MICCAI 2021 Diffusion Simulated Connectivity (DiSCo) Challenge synthetic dataset. The DiSCo dataset are one of a kind numerical phantoms for the simulation of the diffusion-weighted images (DWIs) via Monte-Carlo diffusion simulations. The microscopic and macroscopic complexity of the synthetic substrates allows the evaluation of processing pipelines for the estimation of the quantitative structural connectivity. The diffusion-weighted signal in each voxel of the DWIs is obtained from Monte-Carlo simulations of particle dynamics within a substrate of an unprecedented size of 1 mm(3), allowing for an image matrix size up to 40 x 40 x 40 voxels (isotropic voxel sizes of 25 mu m). In this paper, we provide a characterization of the microstructural properties of the DiSCo dataset, which is composed of three numerical phantoms with comparable microstructure. We report the ground-truth tissue volume fractions ("intra-axonal", "extra-axonal", "myelin"), the fibre density, the bundle density and the fibre orientation distributions (FODs). We believe that this characterization will be beneficial for validating quantitative structural connectivity processing pipelines, and that could eventually find use in microstructural modelling based on machine learning approaches.