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Nested convolutional codes are a set of convolutional codes that is derived from a given generator matrix. The structural properties of nested generator matrices and nested convolutional codes are studied. A method to construct the set of all (rational) minimal generator matrices of a given convolutional code is presented. Nested convolutional codes derived from equivalent minimal generator matrices can have a significant difference in their free distance profiles. This emphasizes the importance of being careful when selecting the generator matrix that determines the nested convolutional codes.