State space realizations and monomial equivalence for convolutional codes

We will study convolutional codes with the help of state space realizations. It will be shown that two such minimal realizations belong to the same code if and only if they are equivalent under the full state feedback group. This result will be used in order to prove that two codes with positive Forney indices are monomially equivalent if and only if they share the same adjacency matrix. The adjacency matrix is an invariant of the code obtained via a minimal state space realization and counts in a detailed way the weights of all possible outputs. It contains full information about the weights of the codewords in the given code.