  Superregular matrices have the property that all of their submatrices, which
can be full rank are so. Lower triangular superregular matrices are useful for
e.g., maximum distance separable convolutional codes as well as for
(sequential) network codes. In this work, we provide an explicit design for all
superregular lower triangular Toeplitz matrices in GF(2^p) for the case of
matrices with dimensions less than or equal to 5 x 5. For higher dimensional
matrices, we present a greedy algorithm that finds a solution provided the
field size is sufficiently high. We also introduce the notions of jointly
superregular and product preserving jointly superregular matrices, and extend
our explicit constructions of superregular matrices to these cases. Jointly
superregular matrices are necessary to achieve optimal decoding capabilities
for the case of codes with a rate lower than 1/2 , and the product preserving
property is necessary for optimal decoding capabilities in network recoding.
