Good Distance Lattices from High Dimensional Expanders
Good Distance Lattices from High Dimensional Expanders
复制标题
高维扩展器的良好距离格子
DOI：
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复制
发表时间：
2018
期刊：
影响因子：
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通讯作者：
David Mass
中科院分区：
文献类型：
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作者：
T. Kaufman;David Mass
我们展示了一种从高维扩展器构造良好距离晶格的新框架。对于纠错码，它和格子有相似的味道，有一种已知的框架，可以从扩展器中产生良好的代码。然而，在我们的工作之前，还没有一个框架可以直接从扩展器中产生良好的距离晶格。有趣的是，我们需要高维展开的概念(而不仅仅是一维展开)来获得密集的大间距晶格。
我们的构造是通过证明在任何环上，特别是在$mathbb{Z}$上的有界度高维余晶扩张子的存在性得到的。以前的有界度共晶扩张器仅在$mathbb{F}_2$以上已知。证明了任意环上的余晶展开由两个主要步骤组成，每个步骤都有各自独立的意义：我们证明了有界度复形的链环上的余界展开意味着该复形是任意环上的余晶扩张子。然后，我们证明了Ramanujan复形(称为球状建筑物)的所有链环都是任意环上的共界扩张子。
我们遵循[LMM16]的策略，证明了球面建筑是任意环上的余边界扩张子。除了将他们的证明从$mathbb{F}_2$推广到任何环之外，我们还详细地介绍了它，这可能会为希望进入该领域的背景较少的读者服务。
We show a new framework for constructing good distance lattices from high dimensional expanders. For error-correcting codes, which have a similar flavor as lattices, there is a known framework that yields good codes from expanders. However, prior to our work, there has been no framework that yields good distance lattices directly from expanders. Interestingly, we need the notion of high dimensional expansion (and not only one dimensional expansion) for obtaining large distance lattices which are dense.
Our construction is obtained by proving the existence of bounded degree high dimensional cosystolic expanders over any ring, and in particular over $mathbb{Z}$. Previous bounded degree cosystolic expanders were known only over $mathbb{F}_2$. The proof of the cosystolic expansion over any ring is composed of two main steps, each of an independent interest: We show that coboundary expansion over any ring of the links of a bounded degree complex implies that the complex is a cosystolic expander over any ring. We then prove that all the links of Ramanujan complexes (which are called spherical buildings) are coboundary expanders over any ring.
We follow the strategy of [LMM16] for proving that the spherical building is a coboundary expander over any ring. Besides of generalizing their proof from $mathbb{F}_2$ to any ring, we present it in a detailed way, which might serve readers with less background who wish to get into the field.
DOI：
10.4171/owr/2016/46
发表时间：
2016
期刊：
Oberwolfach Reports
影响因子：
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作者：
Loeser F
通讯作者：
Loeser F