On superregular matrices and MDP convolutional codes

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.