Abstract:
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, partitions of a space, bounds, bent functions.
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