Extremal subspace covers in finite vector spaces

Abstract

Let V be an n-dimensional vector space over the finite field 𝔽_q, where n≥ 2. It is classical that q+1 proper subspaces are necessary and sufficient to cover V. We study extremal refinements of this covering theorem. For 1≤ m≤ q+1, we determine the maximum possible size of the union of m proper subspaces of V, proving that max_{W₁,…,W_m<V} |W₁∪⋯∪ W_m| = qⁿ⁻²(1+m(q−1)). We also classify all equality cases: for m≥ 2, equality holds precisely when the subspaces are distinct hyperplanes containing a common codimension-two subspace. As consequences, we obtain a structural classification of minimum covers of V by proper subspaces and a sharp defect estimate for unions of q proper subspaces. We then introduce basis-blocking families, namely families of proper subspaces whose union meets every basis of V. We prove that the minimum size of such a family is q and classify all extremal families of this size. Finally, we establish the affine analogue and give an elementary recognition criterion for extremal families. Together, these results provide a unified extremal-combinatorial description of coverings of finite vector spaces by proper subspaces.

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BibTeX RIS

BibTeX

@misc{aliabadi2026extremal,
  title = {{Extremal subspace covers in finite vector spaces}},
  author = {Mohsen Aliabadi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07051},
  eprint = {2610.07051},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07051},
}

RIS

TY  - UNPB
TI  - Extremal subspace covers in finite vector spaces
AU  - Mohsen Aliabadi
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.07051
UR  - https://arxiv.org/abs/2610.07051
AB  - Let V be an n-dimensional vector space over the finite field 𝔽_q, where n≥ 2. It is classical that q+1 proper subspaces are necessary and sufficient to cover V. We study extremal refinements of this covering theorem. For 1≤ m≤ q+1, we determine the maximum possible size of the union of m proper subspaces of V, proving that max_W_1,…,W_m<V |W_1∪⋯∪ W_m| = q^n−2(1+m(q−1)). We also classify all equality cases: for m≥ 2, equality holds precisely when the subspaces are distinct hyperplanes containing a common codimension-two subspace. As consequences, we obtain a structural classification of minimum covers of V by proper subspaces and a sharp defect estimate for unions of q proper subspaces. We then introduce basis-blocking families, namely families of proper subspaces whose union meets every basis of V. We prove that the minimum size of such a family is q and classify all extremal families of this size. Finally, we establish the affine analogue and give an elementary recognition criterion for extremal families. Together, these results provide a unified extremal-combinatorial description of coverings of finite vector spaces by proper subspaces.
ER  -

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