A description of how places split in an asymptotically optimal tower of function fields studied by Garcia and Stichtenoth (1995) is provided and an exact count of the number of places of degree one is given. This information is useful in the setting up of generator matrices for algebraic-geometry codes constructed over this function field tower. These long codes have performance that asymptotically improves upon the Gilbert-Varshamov bound.

Since the proof in 1982, by Tsfasman Vladut and Zink of the existence of algebraic-geometric (AG) codes with asymptotic performance exceeding the Gilbert-Varshamov (G-V) bound, one of the challenges in coding theory has been to provide explicit constructions for these codes. In a major step forward during 1995-1996, Garcia and Stichtenoth (GS) provided an explicit description of algebraic curves, such that AG codes constructed on them would have a performance better than the G-V bound. We present the first low-complexity algorithm for obtaining the generator matrix for AG codes on the curves of GS. The symbol alphabet of the AG code is the finite field of q/sup 2/, q/sup 2//spl ges/49, elements. The complexity of the algorithm, as measured in terms of multiplications and divisions over the finite field GF(q/sup 2/), is upper-bounded by [Nlog/sub q/(N)]/sup 3/ where N is the length of the code. An example of code construction using the above algorithm is presented. By concatenating the AG code with short binary block codes, it is possible to obtain binary codes with asymptotic performance close to the G-V bound. Some examples of such concatenation are included.