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It is shown that each chain of linear codes has an associated demi-matroid, a combinatorial structure that extends the notion of a vector matroid of a linear code. These demi-matroids are proven to determine important properties of the chain, and it is shown that linear code chain duality is represented by demi-matroid duality in a natural and yet surprising way. Profiles are defined for arbitrary demi-matroids and thus for linear code chains, which generalizes previous results in the literature and provides simple and transparent proofs thereof. Finally, applications are given to encryption of messages through wire-tap channels of Type II.