The theory of linear codes over finite fields has long been strengthened by combinatorial theory, and matroid theory has shown to be particularly well suited for providing a theoretical foundation for these codes. Complete descriptions have been given of how important structures such as the (Hamming) supports, weights and higher weights of linear codes over finite fields are determined in combinatorial terms. Through these results, many of the classical theorems from coding theory have now been generalized to matroid theory.
Compared to linear codes over finite fields, combinatorial theory has not yet played the same role for linear codes over other rings. In this talk I will discuss how a class of latroids, a generalization of matroids introduced by D. Vertigan in 2004, seems to be well suited as a theoretical foundation for linear codes over rings and how the theories of latroids and linear codes over rings can be used to enrich both subjects.