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Let V be an n-dimensional vector space over the finite field F q and let V k q denote the family of all k-dimensional subspaces of V. A family F ⊆ V k q is called intersecting if for all F, F ′ ∈ F , we have dim (F ∩ F ′) ≥ 1. A family F ⊆ V k q is called almost intersecting if for every F ∈ F there is at most one element F ′ ∈ F satisfying dim (F ∩ F ′) = 0. In this paper we investigate almost intersecting families in the vector space V. Firstly, for large n, we determine the maximum size of an almost intersecting family in V k q , which is the same as that of an intersecting family. Secondly, we characterize the structures of all maximum almost intersecting families under the condition that they are not intersecting. [ABSTRACT FROM AUTHOR]

A t-spontaneous emission error design, denoted by t-(v, k; m) SEED or t-SEED in short, is a system B of k-subsets of a v-set V with a partition B 1 , B 2 , … , B m of B satisfying | { B ∈ B i : E ⊆ B } | | B i | = μ E for any 1 ≤ i ≤ m and E ⊆ V , | E | ≤ t , where μ E is a constant depending only on E. A t-(v, k; m) SEED is an important combinatorial object with applications in quantum jump codes. The number m is called the dimension of the t-SEED and this corresponds to the number of orthogonal basis states in a quantum jump code. For given v, k and t, a t-(v, k; m) SEED is called optimal when m achieves the largest possible dimension. When k ∣ v , an optimal 1-(v, k; m) SEED has dimension v - 1 k - 1 and can be constructed by Baranyai’s Theorem. This note investigates optimal 1-(v, k; m) SEEDs with k ∤ v , in which a generalization of Baranyai’s Theorem plays a significant role. To be specific, we construct an optimal 1-(v, k; m) SEED for all positive integers v, k, s with v ≡ - s (mod k), k ≥ s + 1 and v ≥ max { 2 k , s (2 k - 1) } . [ABSTRACT FROM AUTHOR]

For distributed storage based on data replication, an optimal data placement is vital to minimize the variance of the number of available files. Well-balanced triple systems and nearly well-balanced triple systems (NWBTSs) were proposed to produce optimal data placements for triple replication. This article concentrates on the existence of NWBTSs. By constructing candelabra systems with various desirable partitions, many new constructions for NWBTSs are developed. The main result of this paper is that there always exist optimal data placements for triple replication on a v-set for all positive integers v possibly except when v ≡ 4 (mod 24) or v = 50 , 74 . [ABSTRACT FROM AUTHOR]