Reference Work Entry

Encyclopedia of Algorithms

pp 434-436

# Learning Heavy Fourier Coefficients of Boolean Functions

1989; Goldreich, Levin
• Luca TrevisanAffiliated withDepartment of Computer Science, University of California at Berkeley

## 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 for the s ...

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