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• W2906838315 abstract "A k-LIN instance is a system of m equations over n variables of the form si1 + · · · + sik = 0 or 1 modulo 2 (each involving k variables). We consider two distributions on instances in which the variables are chosen independently and uniformly but the right-hand sides are different. In a noisy planted instance, the right-hand side is obtained by evaluating the system on a random planted solution and adding independent noise with some constant bias to each equation; whereas in a random instance, the right-hand side is uniformly random. Alekhnovich (FOCS 2003) conjectured that the two are hard to distinguish when k = 3 and m = O(n).We give a sample-efficient reduction from solving noisy planted k-LIN instances (a sparse-equation version of the Learning Parity with Noise problem) to distinguishing them from random instances. Suppose that m-equation, n-variable instances of the two types are efficiently distinguishable with advantage e. Then, we show that O(m · (m/e)2/k)-equation, n-variable noisy planted k-LIN instances are efficiently solvable with probability exp -O((m/e)6/k). Our solver has worse success probability but better sample complexity than Applebaum's (SICOMP 2013). We extend our techniques to show that this can generalize to (possibly non-linear) k-CSPs.The solver is based on a new approximate local list-decoding algorithm for the k-XOR code at large distances. The k-XOR encoding of a function F: Σ → {-1,1} is its k-th tensor power Fk(x1,...,xk) = F(x1)· · · F(xk). Given oracle access to a function G that μ-correlates with Fk, our algorithm, say for constant k, outputs the description of a message that Ω(μ1/k)-correlates with F with probability exp(−O(μ(−4/k)). Previous decoders, for such k, have a worse dependence on μ (Levin, Combinatorica 1987) or do not apply to subconstant μ 1/k. We also prove a new XOR lemma for this parameter regime.The decoder and its analysis rely on a new structure-versus-randomness dichotomy for general Boolean-valued functions over product sets, which may be of independent interest." @default.
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• W2906838315 date "2019-01-06" @default.
• W2906838315 modified "2023-09-26" @default.
• W2906838315 title "XOR codes and sparse learning parity with noise" @default.
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• W2906838315 doi "https://doi.org/10.5555/3310435.3310496" @default.
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