We will discuss our results showing that degree-d group-polynomials are locally correctable with nearly O(log n)^d queries, for the fraction of errors approaching half the minimum distance of the code. Key to this is constructing a novel "interpolating set" for degree-d polynomials, that lies between two exponentially close parallel hyperplanes.
Finally, I will sketch some of the techniques used to demonstrate the list-decodability (in fact, local list-correctability as well) of these codes for error rates approaching their minimum distance. This relies on combining results about the anti-concentration of low-degree polynomials, the sunflower lemma, and the footprint bound for counting common zeros of polynomials.
This is joint work with Amik Raj Behera, Manaswi Paraashar, Srikanth Srinivasan and Madhu Sudan.
