In the field of error-correcting codes, one often wants to work with codes that contain a maximal amount of content and error-correcting data relative to their length. Reed-Solomon codes, which were created in the 1960s, are optimal in this sense.; In the late 1970s, Valeri Goppa introduced algebraic geometry codes, or geometric Goppa codes, which are a generalization of Reed-Solomon codes. The construction of geometric Goppa codes involves points and functions on algebraic curves. Certain Goppa codes gave the first infinite families of codes whose parameters beat the Gilbert-Varshamov bound. With this new link between algebraic geometry and coding theory, mathematicians and computer scientists began to look for towers of curves that give rise to families of codes with a strictly positive percentage of content and error-correcting data. Such towers of curves are called asymptotically good.; For a tower to be asymptotically good, the number of points must be proportional to the genus in each level of the tower. Such towers are surprisingly difficult to find. The existence of asymptotically good towers was known in the 1980s, but it was not until 1995 that explicit examples were found. However, even with these explicit examples, the construction of the corresponding Goppa codes has proved to be difficult.; This dissertation is comprised of four chapters. In the first chapter, I provide preliminaries that combine coding theory, algebraic geometry, and number theory. In the second chapter, I compile a survey of recent papers, outline the methods that reach the limit of current knowledge, and state the open questions. In the third chapter, I study a class of towers of curves for which code construction is relatively straightforward and calculate the genus of each level. Finally, working with a known asymptotically good tower over the finite field with eight elements, I explicitly construct codes from the first three levels and demonstrate a method to construct codes from each subsequent level.
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