  Motivated by a concept studied in [1], we consider a property of matrices
over finite fields that generalizes triangular totally nonsingular matrices to
block matrices. We show that (1) matrices with this property suffice to
construct good tree codes and (2) a random block-triangular matrix over a field
of quadratic size satisfies this property. We will also show that a
generalization of this randomized construction yields codes over quadratic size
fields for which the sum of the rate and minimum relative distance gets
arbitrarily close to 1.
