Explicit Constructions of Asymptotically Good Algebraic - Geometric Codes

在过去的二十年里，纠错码理论中最令人兴奋的发展之一是发现了一类新的线性码，称为代数几何（AG）码。 已知AG码比广泛使用的里德-所罗门码更有效，因为它们在码参数的选择中提供更大的灵活性。 此外，已经证明存在超过Gilbert-Varshamov界的AG码序列;然而，这种AG码的显式构造一直是一个活跃的研究课题，并且仍然是一个开放的问题。 在这个项目中，我们将研究以下问题：（1）研究渐近好的AG码序列的显式构造;（2）研究这些构造以获得超过GF（q^2）上码的Gilbert-Varshamov界的渐近好的AG码，其中q不小于7。

One of the most exciting developments of the last two decades in the theory of error-correcting codes has been the discovery of a new class of linear codes called algebraic-geometric (AG) codes. AG codes are known to be more effective than the widely-used Reed- Solomon codes, as they offer greater flexibility in the choice of code parameters. Moreover, it has been proven that there are sequences of AG codes that exceed the Gilbert-Varshamov bound; however, the explicit construction of such AG codes has been an active research topic and is still an open problem. In this project, the following problems are explored: (1) to investigate the explicit construction of sequences of asymptotically good AG codes; and (2) to investigate these constructions to obtain asymptotically good AG codes that exceed the Gilbert-Varshamov bound for codes over GF(q^2), where q is no less than seven.