A linear ðq; d q; tÞ-perfect hash family of size s consists of a vector space V of order q d over a field F of order q and a sequence 1 ; . . . ; s of linear functions from V to F with the following property: for all t subsets X V, there exists i 2 f1; . . . ; sg such that i is injective when restricted to F. A linear ðq d ; q; tÞ-perfect hash family of minimal size dðt À 1Þ is said to be optimal. In this paper, we prove that optimal linear ðq 2 ; q; 4Þ-perfect hash families exist only for q ¼ 11 and for all prime powers q > 13 and we give constructions for these values of q.

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A linear (q d , q, t)-perfect hash family of size s in a vector space V of order q d over a field F of order q consists of a sequence φ 1 , . . . , φ s of linear functions from V to F with the following property: for all t subsets X ⊆ V there exists i ∈ {1, . . . , s} such that φ i is injective when restricted to F . A linear (q d , q, t)-perfect hash family of minimal size d(t − 1) is said to be optimal. In this paper we use projective geometry techniques to completely determine the values of q for which optimal linear (q 3 , q, 3)-perfect hash families exist and give constructions in these cases. We also give constructions of optimal linear (q 2 , q, 5)-perfect hash families.

In PG(2, q 3 ), let π be a subplane of order q that is exterior to ℓ ∞ . The exterior splash of π is defined to be the set of q 2 + q + 1 points on ℓ ∞ that lie on a line of π. This article investigates properties of an exterior order-q-subplane and its exterior splash. We show that the following objects are projectively equivalent: exterior splashes, covers of the circle geometry CG(3, q), Sherk surfaces of size q 2 +q+1, and scattered linear sets of rank 3. We compare our construction of exterior splashes with the projection construction of a linear set. We give a geometric construction of the two different families of sublines in an exterior splash, and compare them to the known families of sublines in a scattered linear set of rank 3.

In PG(2, q 3 ), let π be a subplane of order q that is tangent to ∞ . The tangent splash of π is defined to be the set of q 2 + 1 points on ∞ that lie on a line of π. This article investigates properties of the tangent splash. We show that all tangent splashes are projectively equivalent, investigate sublines contained in a tangent splash, and consider the structure of a tangent splash in the Bruck-Bose representation of PG(2, q 3 ) in PG(6, q). We show that a tangent splash of PG(1, q 3 ) is a GF(q)-linear set of rank 3 and size q 2 + 1; this allows us to use results about linear sets from [17] to obtain properties of tangent splashes. arXiv:1303.5509v2 [math.CO] 7 Apr 2014 2 The Bruck-Bose representation of PG(2, q 3 ) in PG(6, q) The Bruck-Bose representationWe begin by describing the Bruck-Bose representation of PG(2, q 3 ) in PG(6, q), and introduce the notation we will use.A 2-spread of PG(5, q) is a set of q 3 + 1 planes that partition PG(5, q). A 2-regulus of