We continue the investigation of locally testable codes, i.e., error-correcting codes for whom membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed notion of {\em robust} probabilistically checkable proofs, we introduce the notion of {\em robust} local testability of codes. We relate this notion to a product of codes introduced by Tanner, and show a very simple composition lemma for this notion. Next, we show that codes built by tensor products can be tested robustly and somewhat locally, by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing low-degree multivariate polynomials (which are a special case of tensor codes).
Combining these two results gives us a generic construction of codes of inverse polynomial rate, that are testable with poly-logarithmically many queries. We note these locally testable tensor codes can be obtained from {\em any} linear error correcting code with good distance. Previous results on local testability, albeit much stronger quantitatively, rely heavily on algebraic properties of the underlying codes.
Locally Recoverable Codes with Availability $t\geq 2$ from Fiber Products of Curves
本文利用曲线的纤维积的结构，将Barg，Tamo和Vlaud Duffel给出的代数曲线上的局部可恢复码的构造推广到具有任意多个恢复集的代数曲线上.采用极大曲线，我们创建了几个新的家庭的本地可恢复的代码与多个恢复集，包括代码与两个恢复集的广义Giulietti和Kommarros（GK）曲线和铃木曲线，和新的本地可恢复的代码与许多恢复集的基础上的Hermitian曲线，使用一个纤维积建设的van der Geer和van der Vlugt。此外，我们考虑局部误差恢复和全局误差校正之间的关系，以及本地恢复任何固定数量的擦除模式所需的可用性。
