BSF:2014359:Invariance in error correcting codes and computational complexity

可以对信息进行数字编码(如0和1)，以便即使这些比特中的某些比特丢失或修改，也可以恢复信息。这一基本思想是DVD和数字广播等常见技术的基础，在理论计算机科学中进行了探索，在概率可核查证明和性质测试中发挥了关键作用。在这些抽象的环境中探索这一想法的局限性可以带来新的理解，从而使技术创新成为可能。在数学中，研究对象的不变性(对称性)对于理解对象具有重要的作用，这在数学中由来已久。近年来，不变性的研究在理论计算机科学的各个领域取得了重大进展，如性质测试、纠错码和复杂性理论。例如，具有仿射不变性的码和具有传递不变群的码在复杂性理论中有重要的应用，例如构造PCP。该项目将支持私人投资及其学生与以色列的合作者就这些主题进行研究合作的旅行。这项研究将加深我们对具有不变性的对象的理解，并开拓这一理论在复杂性理论中的新应用。

Information can be digitally encoded (as 0s and 1s) so that even if some of these bits are lost or modified, the information can be recovered. This foundational idea, which is the basis for such commonplace technologies as DVDs and digital broadcast, is explored in theoretical computer science, where it plays a key role in probabilistically checkable proofs and property testing. Exploring the limits of this idea in these abstract settings can lead to new understanding, which enables technological innovation. It has been long known in mathematics that studying the invariances (symmetries) of an object has an important role in understanding the object. In recent years, the study of invariances has led to some significant advances in various topics in theoretical computer science, such as property testing, error-correcting codes and complexity theory. For example, codes with affine invariance and codes with a transitive invariance group have been shown to have important applications in complexity theory, such as constructing PCPs. This project will support the travel of the PIs and their students for research collaborations on these topics with collaborators in Israel. This research will deepen our understanding of objects with invariances, and develop new applications of this theory in complexity theory.