Ryan Hutchinson

Superregular matrices are a type of lower triangular Toeplitz matrix that arises in the context of constructing convolutional codes having a maximum distance profile. These matrices are characterized by the property that the only submatrices having a zero determinant are those whose determinants are trivially zero due to the lower triangular structure. In this paper, we discuss how superregular matrices may be used to construct codes having a maximum distance profile. We also present an upper bound on the minimum size a finite field must have in order that a superregular matrix of a given size can exist over that field. This, in turn, gives an upper bound on the smallest field size over which an MDP (n,k,δ) convolutional code can exist.

Maximum distance profile codes are characterized by the property that two trajectories which start at the same state and proceed to a different state will have the maximum possible minimum distance from each other relative to any other convolutional code of the same rate and degree. In this paper we use methods from systems theory to characterize maximum distance profile codes algebraically. The main result shows that maximum distance profile codes form a generic set inside the variety which parametrizes the set of convolutional codes of a fixed rate and a fixed degree.