Purity testing protocols (PTPs), also known as Bell state testing protocols, i.e., protocols that decide with high probability whether or not a distributed bipartite quantum state is a tensor product of Bell states, have been proven to be a useful tool in many quantum communication applications. In this paper, we provide geometrical constructions for such protocols that originate directly from classical linear error correcting codes (LECCs), in a way that the properties of the resulting PTP are completely determined from those of the LECC used in the construction. We investigate the implications of our results in various quantum communication tasks, including error detection, entanglement purification for general quantum error models and quantum message authentication.