@MISC{And_aconstant, author = {Lecture May And}, title = {A Constant Number of Query Bits}, year = {} }

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Abstract

rd to z, and then we may think of x as the decoding of z. There are two dual views of the Hadamard code, based on two different interpretations of i=1 x i y i . 1. View the x i as coefficients and the y i as variables. Then the codeword E(x) can be viewed as the linear function f x = i=1 x i y i evaluated on all possible inputs. 2. View the y i as coefficients and the x i as variables. Then the codeword E(x) can be viewed as evaluating all possible linear functions (over GF (2) ) at the point x 2 f0; 1g . 1.2 The linearity test Given a string z 2 f0; 1g , we would like to test whether it is (close to) a codeword of the Hadamard code. As noted in Section 1.1, valid codewords can be viewed as linear functions f x . Likewise, we view z as a Boolean function f , and accessing the bit at location y 2 f0; 1g can be viewed as getting the value of f(y). For two strings x; y 2 f0; 1g , let x \Phi y 2 f0; 1g denote their bitwise exclusive or. The linearity test: Choose