Algebraic geometry provides methods for finding families of function fields having an asymptotically good behaviour for coding-theoretical applications. They lead to error-correction codes of long length and good performance. The performance depends on the ratio \(N/g\) of the genus \(g\) of the function field and the number \(N\) of its places over the finite field. The Drinfeld-Vladut bound \(\lim_{g\to\infty}\sup N/g\leq \sqrt{q}-1\) gives an upper bound on the asymptotic value of this ratio. M. A. Tsfasman, S. G. Vladut, and Th. Zink [Math. Nachr. 109, 21-28 (1982; Zbl 0574.94013)] proved the existence of families attaining this bound. A. Garcia and H. Stichtenoth [J. Number Theory 61, 248-273 (1996; Zbl 0893.11047)] managed to state two families of curves in an explicit way which also achieve this bound. Although this description is far more explicit, to give the corresponding AG code one needs to compute a basis of the vector space \(\mathcal{L}(rP)\), where \(P\) is a place in the function field and \(r\) is an integer.
The authors give integral bases for \(\mathcal{O}_{T_n}\) over \(\mathcal{O}_{T_1}\), where the \(T_i\) denote the fields in the second tower of function fields given by Garcia and Stichtenoth. They first give a set that forms a local integral basis at all places except for \(P_\infty\) and \(P_0\), then they only exclude \(P_0\). These sets can now be used to give the AG codes.
For the entire collection see [Zbl 0959.00027].