高维扩展意味着放大的局部可测性
High dimensional expansion implies amplified local testability
摘要：
在这项工作中，我们表明，高维扩展意味着本地可测试的代码。具体来说，我们定义了一个概念，我们称之为高维扩展系统（HDE系统）。这是一个由关联关系定义的集合系统，其集合之间具有一定的高维展开关系。如果一个线性代码所满足的线性约束的集合可以通过HDE-系统来描述，那么我们称这个线性代码是在HDE-系统上建模的。我们证明了一个可以在HDE系统上建模的代码是局部可测试的。这意味着高维扩展现象只意味着代码的局部可测性。以前的工作必须依靠当地的概念，当地的可测试性，以获得一些全球形式的可测试性（例如，从当地的共收缩膨胀，从当地的全球协议），而我们的工作推断全球的可测试性直接从高维扩展，而不依赖于一些当地形式的可测试性。我们从HDE-系统得到的局部可测性结果实际上比标准的强，我们称之为放大的局部可测性。我们进一步表明，大多数研究良好的局部可测试码的Reed-Muller码和更一般的仿射不变码与单轨道属性属于我们的框架。也就是说，可以证明它们是在HDE系统上建模的，因此所有p元仿射不变码的家族都是局部可测试的。这产生了最强的已知测试结果的仿射不变码与单轨道，加强了考夫曼和苏丹的工作。
英文摘要：
In this work we show that high dimensional expansion implies locally testable code. Specifically, we define a notion that we call high-dimensional-expanding-system (HDE-system). This is a set system defined by incidence relations with certain high dimensional expansion relations between its sets. We say that a linear code is modelled over HDE-system, if the collection of linear constraints that the code satisfies could by described via the HDE-system. We show that a code that can be modelled over HDE-system is locally testable. This implies that high dimensional expansion phenomenon solely implies local testability of codes. Prior work had to rely to local notions of local testability to get some global forms of testability (e.g. co-systolic expansion from local one, global agreement from local one), while our work infers global testability directly from high dimensional expansion without relying on some local form of testability. The local testability result that we obtain from HDE-systems is, in fact, stronger than standard one, and we term it amplified local testability. We further show that most of the well studied locally testable codes as Reed-Muller codes and more generally affine invariant codes with single-orbit property fall into our framework. Namely, it is possible to show that they are modelled over an HDE-system, and hence the family of all p-ary affine invariant codes is amplified locally testable. This yields the strongest known testing results for affine invariant codes with single orbit, strengthening the work of Kaufman and Sudan.
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