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.

翻译：环面码由 Hansen 引入，作为 Reed-Solomon 码的推广，是 $\mathbb{F}_q^n$ 中的一个 $k$ 维子空间，它由一个环面簇或其关联的整数凸多胞形 $P \subseteq [0,q-2]^n$ 所决定，其中 $k=|P \cap \mathbb{Z}^n|$（即 $P$ 的整数格点数目）。衡量一个码的质量有两个关键参数：信息率（衡量每个码字单比特所含信息量）与相对最小距离（衡量相对于每个码字的比特数能纠正多少错误）。Soprunov 与 Soprunova 将“优良的无限码族”定义为多胞形维数无界的一列码，使得对应的信息率与相对最小距离在极限下均不趋于 0。本文通过考虑多胞形的联接（join）与直和（direct sum）等运算，探讨了构造码族的不同方式。在此过程中，我们给出了不存在优良码族的条件，并提供了强有力的证据表明此类优良码族并不存在。