  A trivially zero minor of a matrix is a minor having all its terms in the
Leibniz formula equal to zero. A matrix is superregular if all of its minors
that are not trivially zero are nonzero. In the area of Coding Theory,
superregular matrices over finite fields are connected with codes with optimum
distance proprieties. When a superregular matrix has all its entries nonzero,
it is called full superregular and these matrices are used to construct Maximum
Distance Separable block codes. In the context of convolutional codes, lower
triangular Toeplitz superregular matrices are employed to build convolutional
codes with optimal column distance. Although full superregular matrices over
small fields are known (e.g. Cauchy matrices), the few known general
constructions of these matrices having a lower triangular Toeplitz structure
require very large field sizes. In this work we investigate lower triangular
Toeplitz superregular matrices over small finite prime fields. Following the
work of Hutchinson, Smarandache and Trumpf, we study the minimum number of
different nontrivial minors that such a matrix have, and exhibit concrete
constructions of superregular matrices of this kind.
