Learning Heavy Fourier Coefficients of Boolean Functions

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Keywords and Synonyms

Error-control codes, Reed–Muller code     

Problem Definition

The Hamming distance dH(y, z) between two binary strings y and z of the same length is the number of entries in which y and z disagree. A binary error-correcting code of minimum distance d is a mapping \( { C:\{0,1\}^k \rightarrow \{0,1\}^n } \) such that for every two distinct inputs \( { x,x^\prime\in \{0,1\}^k } \), the encodings C(x) and \( { C(x^\prime) } \) have Hamming distance at least d. Error-correcting codes are employed to transmit information over noisy channels. If a sender transmits an encoding C(x) of a message x via a noisy channel, and the recipient receives a corrupt bit string \( { y\neq C(x) } \), then, provided that y differs from C(x) in at most \( { (d-1)/2 } \) locations, the recipient can recover y from C(x). The recipient can do so by searching f ...