Bounds on the number of partitions of the vector space over a finite field into affine subspaces of the same dimension

I. P. Baksova ; Yu. V. Tarannikov

Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 5-8

Résumé

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*}

Détail

Keywords: affine subspaces, bounds, bent functions.
Mots-clés : partitions of a space