基于Galois扩张的折叠代数几何码
Folded Algebraic Geometric Codes From Galois Extensions
摘要：
本文基于函数域的伽罗瓦扩张描述了一类新的列表可译码码，并给出了一个列表译码算法。这些代码是通过使用扩展的伽罗瓦群中的某些元素（自同构）折叠函数域的有理位置集而获得的。本文的工作是将折叠Reed Solomon码推广到代数几何码上。我们描述了两个结构的基础上，这个框架取决于如果用于折叠代码的自同构的顺序是大或小相比，块的长度。当自同构为高阶时，在最坏情况下，码的列表长度多项式有界。这种结构给出了代码的速率$R$在一个字母表的大小独立的块长度，可以纠正一小部分的$1-R-ε$的错误受到存在的渐近良好的塔的功能领域与大的自同构。第二种构造解决了当用于折叠的元素的阶数与块长度相比很小时的情况。在这种情况下，一个启发式分析表明，对于一个随机接收的字，预期的列表大小和解码算法的运行时间是有界的块长度的多项式。当应用于Garcia-Stichtenoth塔时，这在大小为$(\frac{1}{ε^2})^{O(\frac{1}ε)}$的字母表上产生速率为$R$的代码，其可以校正$1-R-ε$错误的一部分。
英文摘要：
We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. These codes are obtained as a result of folding the set of rational places of a function field using certain elements (automorphisms) from the Galois group of the extension. This work is an extension of Folded Reed Solomon codes to the setting of Algebraic Geometric codes. We describe two constructions based on this framework depending on if the order of the automorphism used to fold the code is large or small compared to the block length. When the automorphism is of large order, the codes have polynomially bounded list size in the worst case. This construction gives codes of rate $R$ over an alphabet of size independent of block length that can correct a fraction of $1-R-ε$ errors subject to the existence of asymptotically good towers of function fields with large automorphisms. The second construction addresses the case when the order of the element used to fold is small compared to the block length. In this case a heuristic analysis shows that for a random received word, the expected list size and the running time of the decoding algorithm are bounded by a polynomial in the block length. When applied to the Garcia-Stichtenoth tower, this yields codes of rate $R$ over an alphabet of size $(\frac{1}{ε^2})^{O(\frac{1}ε)}$, that can correct a fraction of $1-R-ε$ errors.
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