The rank r matrix completion problem studies whether a matrix where some of the entries have been filled in with generic complex numbers can be completed to a matrix of rank at most r. This problem is governed by the bipartite rigidity matroid. We show that the study of the bipartite rigidity matroid is related to the study of tensor codes, a topic in information theory, and use this relation to understand new cases of both problems. Joint work with Joshua Brakensiek, Manik Dhar, Jiyang Gao, and Sivakanth Gopi.