Abstract A framework for encoding the boundary of any arbitrary binary shape in a high dimensional digital image is described, with special attention to the 3D case. The voxels of the outer shell of the shape are treated as vertices in a connected subgraph of an undirected cubic grid graph which has at least a spanning tree. A traversal of this spanning tree visits each voxel in a repeatable order. The sequence of traversed edges is coded into a chain that constitutes a high dimensional chain code, analogous to the classical Freeman chain code for 2D. The outer surface of the shape can be unambiguously reconstructed from this chain code. The framework is general in the sense that it can be used under any connectivity rule in any practical number of dimensions and also with noncubic voxels. Explicit algorithms for the computation of the chain codes and the reconstruction of the digital shapes in 3D and 4D are detailed. These are based on a hybrid graph traversal algorithm. The algorithms are illustrated with some simple digital shapes and tested with a benchmark dataset of medical images of intervertebral discs.