Motivated by applications in reliable and secure communication, we address the problem of tiling (or partitioning) a finite constellation in $\mathbb{Z}_{2^L}^n$ by subsets, in the case that the constellation does not possess an abelian group structure. The property that we do require is that the constellation is generated by a linear code through an injective mapping. The intrinsic relation between the code and the constellation provides a sufficient condition for a tiling to exist. We also present a necessary condition. Inspired by a result in group theory, we discuss results on tiling for the particular case when the finer constellation is an abelian group as well.
翻译:以可靠和安全通信的应用为动力,我们解决了按子集对一个固定星座用$\mathbb ⁇ ⁇ 2 ⁇ L ⁇ n$(或分割)打砖(或分割)的问题,如果该星座不拥有贝氏集团结构的话。我们确实需要的属性是,星座是由直线代码通过一个注入式绘图生成的。代码和星座之间的内在关系为铺砖的存在提供了充分的条件。我们还提出了一个必要条件。根据集体理论的结果,我们讨论当精细星座是一个贝氏集团时,特定案件的打砖结果。