论文标题
关于良好的无限型福利代码家庭或缺乏的家族
On Good Infinite Families of Toric Codes or the Lack Thereof
论文作者
论文摘要
汉森（Hansen）介绍的曲折代码将芦苇 - 固体代码扩展为$ \ sumbb {f} _q^n $的$ k $维度，取决于感谢您或其相关的积分convex polytope $ p polytope $ p \ subseteq [0，q-2]^n $ k = p \ p \ p \ p \ z |整数晶格点$ p $）。有两个相关参数可以确定代码的质量：信息速率，该参数衡量了每个代码字的一位中包含多少信息；以及相对最小距离，该距离可以衡量相对于每个代码字数量有多少位可以纠正多少误差。 Soprunov和Soprunova将良好的无限代码家族定义为一系列无界的多层尺寸代码，因此相应的信息速率或相对最小距离都不在极限中达到0。我们通过考虑诸如联接和直接总和之类的多层操作来研究不同的代码家族的方式。通过这样做，我们提供了没有一个好家庭可以存在的条件，并且有充分的证据表明没有那么好的代码家庭。
A toric code, introduced by Hansen to extend the Reed-Solomon code as a $k$-dimensional subspace of $\mathbb{F}_q^n$, is determined by a toric variety or its associated integral convex polytope $P \subseteq [0,q-2]^n$, where $k=|P \cap \mathbb{Z}^n|$ (the number of integer lattice points of $P$). There are two relevant parameters that determine the quality of a code: the information rate, which measures how much information is contained in a single bit of each codeword; and the relative minimum distance, which measures how many errors can be corrected relative to how many bits each codeword has. Soprunov and Soprunova defined a good infinite family of codes to be a sequence of codes of unbounded polytope dimension such that neither the corresponding information rates nor relative minimum distances go to 0 in the limit. We examine different ways of constructing families of codes by considering polytope operations such as the join and direct sum. In doing so, we give conditions under which no good family can exist and strong evidence that there is no such good family of codes.
