Year: 2026
Type: article
Abstract: Let $$V=V(d,q)$$ V = V ( d , q ) denote the vector space of dimension d over $${\mathbb F}_q$$ F q . A subspace partition $$\mathcal {P}$$ P of V , also known as a vector space partition , is a collection of nonempty subspaces of V such that each nonzero vector of V is in exactly one subspace of $$\mathcal {P}$$ P . Motivated by applications of minimum blocking sets and maximal partial t-spreads , Beutelspacher (Geom Dedic 9:425–449, 1980) determined in a lemma the minimum possible size $$\delta (d)$$ δ ( d ) over all (nontrivial) subspace partitions of V . In Heden et al. (Des Codes Cryptogr 64:265–274, 2012) and Năstase and Sissokho (Linear Algebra Appl 435:1213–1221, 2011), we extended Beutelspacher’s Lemma by determining the (first) minimum size $$\sigma _q(d,t)$$ σ q ( d , t ) of any subspace partition of V for which the largest subspace has dimension t , with $$1\le t 1 ≤ t < d . In this paper, we build on the previous results and unveil additional structural information of subspace partitions. We determine the second minimum size $$\delta '(d)$$ δ ′ ( d ) over all (nontrivial) subspace partitions of V and furthermore, for $$d\equiv r \pmod {t}$$ d ≡ r ( mod t ) and $$0\le r 0 ≤ r < t < d , we prove the exact value of the second minimum size $$\sigma _q'(d,t)$$ σ q ′ ( d , t )
Source: Designs Codes and Cryptography
Authors: Esmeralda Năstase, Papa Sissokho
Institutions: Xavier University, Illinois State University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Funder: Illinois State University
OpenAlex ID: W7130348987