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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 convolutional codes and nested generator matrices are studied. A method to construct the set of all minimal (rational) generator matrices of a given convolutional code is presented. As an example, two different sets of nested convolutional codes are derived from two equivalent minimal generator matrices. The significant difference in their free-distance profiles emphasizes the importance of being careful when selecting the generator matrices that determine the nested convolutional codes. As an application of nested convolutional codes, woven codes with outer warp, and inner nested convolutional codes are considered. The free-distance profile of the inner generator matrix is shown to be an important design tool.