Locally decodable codes: a brief survey
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
SIAM Journal on Computing
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Sparse multivariate function recovery from values with noise and outlier errors
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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Let f in F_q[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 2 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; _n]q code to a randomly chosen set of O(k) coordinates with highprobability 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 querycomplexity in order to tolerate a constant fraction of errors. This result yields a moderate reduction in thequery complexity of the currently best known codes. _ We improve the state of the art in constructions of explicit rigid matrices. For any prime power q and integers n and d we construct an explicit matrix M with exp(d) _ n rows and n columns such that the rank of M stays above n=2 even if every row of M is arbitrarily altered in up to d coordinates. Earlier, such constructions were available only for q = O(1) or q = (n):