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Highly nonlinear functions are used in combination generators and other stream ciphers.

We know that maximum nonlinearity (hamming distance to all affine functions) of $f$ is achieved when $f$ is bent function. This will be $$NL=2^{n-1}-2^{(n/2)-1}$$ when $n$ is even. Also when $n$ is odd, this bound is not satisfied. What will it be?

The question is: Can we find $f$ with largest nonlinearity for an arbitrary odd integer $n$ case?

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This is a famous open problem in coding theory, the covering radius of the first order Reed-Muller code, when the number of variables is odd.

As far as I know the best general upper bound on nonlinearity which in this paper is denoted N (=covering radius R) for $n=2k+1$ is still the one below:

enter image description here

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    $\begingroup$ Since the question (quickly deleted) was relevant to crypto and not asked before, I resurrected it. $\endgroup$ – kodlu May 23 at 23:38

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