Noisy Interpolation of Sparse Polynomials, and Applications

Let f ∈ Fq[x] be a polynomial of degree d ≤ q/2. It is well-known that f can be uniquely recovered from its values at some 2d points even after some small fraction of the values are corrupted. In this paper we establish a similar result for sparse polynomials. We show that a k-sparse polynomial f ∈ Fq[x] of degree d ≤ q/2 can be recovered from its values at O(k) randomly chosen points, even if a small fraction of the values of f are adversarially corrupted. Our proof relies on an iterative technique for analyzing the rank of a random minor of a matrix. We use the same technique to establish a collection of other results. Specifically, • We show that restricting any linear [n, k, δn]q code to a randomly chosen set of O(k) coordinates with high probability yields an asymptotically good code. • We improve the state of the art in locally decodable codes, showing that similarly to Reed Muller codes matching vector codes require only a constant increase in query complexity in ...

• Type: Journal
• Year: 2011
• Where: COCO
• Authors: Shubhangi Saraf, Sergey Yekhanin
• Keywords: Algorithms, COCO 2011, Query Complexity, Sparse Polynomial, Sparse Polynomials