Geodesic
On tightness of the lower bound for the number of bent functions at the minimum distance from a bent function from the Maiorana ---McFarland class
Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 14-18

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The lower bound $2^{2n+1} - 2^n$ for the number of bent functions at the minimum distance from a bent function from the Maiorana — McFarland class $\mathcal{M}_{2n}$ in $2n$ variables is investigated. A criterion for the reachability of this lower bound for functions in algebraic representation is presented. It is constructively proven that it is accurate for $n = p^k$, where $p \neq 2,3$ is prime and $k$ is natural. It is shown that a necessary condition for the reachability of the bound is the construction of a function from $\mathcal{M}_{2n}$ using an APN permutation whose set of values on any affine subspace of dimension $3$ is not an affine subspace.
Keywords: bent function, Boolean function, Maiorana — McFarland class, lower bound.
Mots-clés : minimum distance 
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     author = {D. A. Bykov},
     title = {On tightness of the lower bound for the number of bent functions at the minimum distance from a bent function from the {Maiorana} {---McFarland} class},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {14--18},
     publisher = {mathdoc},
     number = {16},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2023_16_a3/}
}
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D. A. Bykov. On tightness of the lower bound for the number of bent functions at the minimum distance from a bent function from the Maiorana ---McFarland class. Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 14-18. http://geodesic.mathdoc.fr/item/PDMA_2023_16_a3/