Document Type

Article

Publication Title

Designs, Codes and Cryptography

Publication Date

2026

Keywords

subspace partition, vector space partition, supertail of a subspace partition, blocking sets, partial spreads, combinatorial stability

Abstract

Let V = V (d, q) denote the vector space of dimension d over Fq . A subspace partition 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 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 δ(d) over all (nontrivial) subspace partitions of V . In Heden et al. (Des Codes Cryptogr 64:265–274, 2012) and N˘astase and Sissokho (Linear Algebra Appl 435:1213–1221, 2011), we extended Beutelspacher’s Lemma by determining the (first) minimum size σq (d, t) of any subspace partition of V for which the largest subspace has dimension t, with 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 δ′(d) over all (nontrivial) subspace partitions of V and furthermore, for d ≡ r (mod t) and 0 ≤ r < t < d, we prove the exact value of the second minimum size σ ′ q (d, t) of any subspace partition of V for which the largest subspace has dimension t and when at least one of the following holds: (i) r = 0, (ii) t + r is even, (iii) d < 2t or (iv) the partition has only subspaces of two different dimensions. Finally, applications to the supertail of a subspace partition and the size of maximal partial spreads are given.

Funding Source

This article was published Open Access thanks to a transformative agreement between Milner Library and Springer Nature.

Comments

First published in Designs, Codes and Cryptography (2026): https://doi.org/10.1007/s10623-025-01753-2

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

DOI

10.1007/s10623-025-01753-2

Download

Included in

Mathematics Commons

Share

COinS