代数几何码的困难问题
Hard Problems of Algebraic Geometry Codes
摘要：
最小距离是码的最重要的组合特征之一。最大似然译码问题是码的最重要的算法问题之一。虽然这些问题对于一般的线性码来说是困难的，但是用来证明其困难性的技术通常依赖于人工码的构造。一般来说，对特定类别的自然线性码的硬度知之甚少。在本文中，我们证明了这两个问题是NP-困难的代数几何码。我们实现这一点，减少了一个众所周知的NP完全问题，这些问题使用随机算法。在减少的代码的家庭是基于椭圆曲线。它们有正的速率，但字母表的大小是块长度的指数。
英文摘要：
The minimum distance is one of the most important combinatorial characterizations of a code. The maximum likelihood decoding problem is one of the most important algorithmic problems of a code. While these problems are known to be hard for general linear codes, the techniques used to prove their hardness often rely on the construction of artificial codes. In general, much less is known about the hardness of the specific classes of natural linear codes. In this paper, we show that both problems are NP-hard for algebraic geometry codes. We achieve this by reducing a well-known NP-complete problem to these problems using a randomized algorithm. The family of codes in the reductions are based on elliptic curves. They have positive rates, but the alphabet sizes are exponential in the block lengths.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
