Geodesic
Bounds on the number of partitions of the vector space over a finite field into affine subspaces of the same dimension
Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 5-8 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We give lower and upper bounds on the number of ordered $N_m^k\left(q\right)$ and unordered $\widetilde{N}_m^k\left(q\right)$ partitions of the space $\mathbb{F}_q^m$ into affine subspaces of the same dimension $k$. In particular, the asymptotics of the logarithm of the number of unordered partitions of the space $\mathbb{F}_3^m$ into one-dimensional affine subspaces is established: $$\dfrac{m}{3}\cdot 3^{m}+c_{1}\cdot 3^{m}+o\left(3^{m}\right)\leq \log_{3}\widetilde{N}^{1}_{m}\left(3\right)\leq \dfrac{m}{3}\cdot 3^{m}+c_{2}\cdot 3^{m}+o\left(3^{m}\right).$$ Also, we highlight the bounds \begin{gather*} \log_q{N_{m}^{k}\left(q\right)}\gtrsim (m-k)q^{m-k}, m-k\rightarrow\infty,\\ \log_3{N_{m}^{k}\left(3\right)}\gtrsim 2\left(m-k\right) 3^{m-k},\\ \log_q N_{m}^{k}\left(q\right)\gtrsim \left(m-\frac{q-1}{q} k\right)q^{m-k}, k\rightarrow\infty, m-k\rightarrow\infty\\ \log_q{N_{m}^{k}\left(q\right)}\leq (k+1)(m-k-\log_q e)q^{m-k}+O(q^{m-k})+O(k(m-k)). \end{gather*}
Keywords: affine subspaces, bounds, bent functions.
Mots-clés : partitions of a space 
@article{PDMA_2023_16_a0,
     author = {I. P. Baksova and Yu. V. Tarannikov},
     title = {Bounds on the number of partitions of the vector space over a finite field into affine subspaces of the same dimension},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {5--8},
     year = {2023},
     number = {16},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2023_16_a0/}
}
TY  - JOUR
AU  - I. P. Baksova
AU  - Yu. V. Tarannikov
TI  - Bounds on the number of partitions of the vector space over a finite field into affine subspaces of the same dimension
JO  - Prikladnaya Diskretnaya Matematika. Supplement
PY  - 2023
SP  - 5
EP  - 8
IS  - 16
UR  - http://geodesic.mathdoc.fr/item/PDMA_2023_16_a0/
LA  - ru
ID  - PDMA_2023_16_a0
ER  - 
%0 Journal Article
%A I. P. Baksova
%A Yu. V. Tarannikov
%T Bounds on the number of partitions of the vector space over a finite field into affine subspaces of the same dimension
%J Prikladnaya Diskretnaya Matematika. Supplement
%D 2023
%P 5-8
%N 16
%U http://geodesic.mathdoc.fr/item/PDMA_2023_16_a0/
%G ru
%F PDMA_2023_16_a0
I. P. Baksova; Yu. V. Tarannikov. Bounds on the number of partitions of the vector space over a finite field into affine subspaces of the same dimension. Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 5-8. http://geodesic.mathdoc.fr/item/PDMA_2023_16_a0/

[1] Agievich S., “Bent rectangles”, NATO Science for Peace and Security Series — D: Information and Communication Security, 18 (2008), 3–22 | MR | Zbl

[2] Baksova I. P., Tarannikov Yu. V., “Ob odnoi konstruktsii bent-funktsii”, Obozrenie priklad. i promyshl. matem., 27:1 (2020), 64–66

[3] Baksova I. P., Tarannikov Yu. V., “Otsenki chisla razbienii prostranstva $\mathbb{F}^m_2$ na affinnye podprostranstva razmernosti $k$”, Vestnik Mosk. un-ta. Ser. 1. Matematika, mekhanika, 2022, no. 3, 21–25 | MR | Zbl

[4] Tarannikov Yu. V., “O suschestvovanii razbienii, primitivnykh po Agievichu”, Diskretnyi analiz i issledovanie operatsii, 29:4 (2022), 125–144 | MR

[5] Potapov V. N., Taranenko A. A., and Tarannikov Yu. V., Asymptotic Bounds on Numbers of Bent Functions and Partitions of the Boolean Hypercube into Linear and Affine Subspaces, 2021, arXiv: 2108.00232

[6] Eberhard S., More on Additive Triples of Bijections, 2017, arXiv: 1704.02407