Many combinatorial problems can be formulated as special cases of the following question: Let $1 \leq k <d$, and let $U_1, … ,U_m$ be $k$-dimensional subspaces of $V := \mathbb{F}_q^d$, where $q$ is a prime power and $\mathbb{F}_q$ is a field with $q$ elements.

A system of representatives for $V/U_i$ is a subset $Q \subset V$ with a single element in each equivalence class $v+U_i$. The question is: How many such sets $Q$ are there that are simultaneously systems of representatives for all of the spaces $V/U_i$, where $1 \leq i \leq m$?

Is there any theory for this general problem? Is it a known or studied problem in Algebra or Algebraic combinatorics?


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