Bezerra, Garcia and Stichtenoth constructed an explicit tower of function fields over a cubic finite field, whose limit attains the Zink bound. Their proof is rather long and very technical. The main aim of this thesis is to replace the complex calculations in their work by structural arguments, thus giving a much simpler and more transparent proof for the limit of the Bezerra–Garcia–Stichtenoth tower. We also compute the limit of the Galois closure of this tower. One of the main tools used while determining the limits of these towers is a lemma from ramification theory. Using the theory of higher ramification groups, we give proof of this result, which is valid for more general fields. Furthermore, using a variant of these towers, we obtain asymptotic lower bounds for the class of r-quasi transitive codes over cubic finite fields and the class of transitive isoorthogonal codes over cubic finite fields.