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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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/

[1] Tokareva N., Bent Functions: Results and Applications to Cryptography, Academic Press, 2015, 220 pp. | MR | Zbl

[2] Rothaus O., “On bent functions”, J. Comb. Theory. Ser. A, 20:3 (1976), 300–305 | DOI | MR | Zbl

[3] Kolomeets N. A., Pavlov A. V., “Svoistva bent-funktsii, nakhodyaschikhsya na minimalnom rasstoyanii drug ot druga”, Prikladnaya diskretnaya matematika, 2009, no. 4(6), 5–20

[4] Kolomeets N. A., “Perechislenie bent-funktsii na minimalnom rasstoyanii ot kvadratichnoi bent-funktsii”, Diskretn. analiz i issled. oper., 19:1 (2012), 41–58 | MR | Zbl

[5] Kolomeets N. A., “Verkhnyaya otsenka chisla bent-funktsii na rasstoyanii $2^k$ ot proizvolnoi bent-funktsii ot $2k$ peremennykh”, Prikladnaya diskretnaya matematika, 2014, no. 3(25), 28–39

[6] Kolomeec N., “The graph of minimal distances of bent functions and its properties”, Des. Codes Cryptogr., 85:3 (2017), 395–410 | DOI | MR | Zbl

[7] Bykov D. A., “O nizhnei otsenke chisla bent-funktsii na minimalnom rasstoyanii ot bent-funktsii iz klassa Meiorana — MakFarlanda”, Prikladnaya diskretnaya matematika. Prilozhenie, 2022, no. 15, 22–25

[8] Logachev O. A., Salnikov A. A., Smyshlyaev S. V., Yaschenko V. V., Bulevy funktsii v teorii kodirovaniya i kriptologii, MTsNMO, M., 2012, 584 pp.