Magnetic particle imaging (MPI) is a relatively new medical imaging modality, which detects the nonlinear response of magnetic nanoparticles (MNPs) that are exposed to external magnetic fields. The system matrix (SM) method for MPI image reconstruction requires a time consuming system calibration scan prior to image acquisition, where a single MNP sample is measured at each voxel position in the field-of-view (FOV). The scanned sample has the maximum size of a voxel so that the calibration measurements have relatively poor signal-to-noise ratio (SNR). In this paper, we present the coded calibration scene (CCS) framework, where we place multiple MNP samples inside the FOV in a random or pseudo-random fashion. Taking advantage of the sparsity of the SM, we reconstruct the SM by solving a convex optimization problem with alternating direction method of multipliers using CCS measurements. We analyze the effects of filling rate, number of measurements, and SNR on the SM reconstruction using simulations and demonstrate different implementations of CCS for practical realization. We also compare the imaging performance of the proposed framework with that of a standard compressed sensing SM reconstruction that utilizes a subset of calibration measurements from a single MNP sample. The results show that CCS significantly reduces calibration time while increasing both the SM reconstruction and image reconstruction performances.