Translations of the squares in a finite field and an infinite family of 3-designs

  • Authors:

  • Shiro Iwasaki

  • Affiliations:

  • Department of Mathematics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2003

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Abstract

Let q be a prime power with q ≡ -1 (mod 4), and let F be the finite field with q elements and Q the set of nonzero squares in F. Let G = PSL(2,q) be the special linear fractional group on Ω = {∞} ∪ F, the projective line over F, and set V = {∞} ∪ (QΔ(Q + 1)Δ(Q - 1)), oV = Ω\V, where Δ denotes the symmetric difference. First, we consider the cardinality of intersections of some translations of Q in F and show |Q ∩ (Q + 1) ∩ (Q - 1)| = {(q - 7)/8 if 2 ∈ Q, (q - 3)/8 otherwise. Next, when 2 ∉ Q, we determine the structure of GV = GoV, the setwise stabilizer of V or oV in G, and show that the design (Ω, oVG) is a 3-(q + 1, (q - 3)/2,λ) design, where λ = {(q - 3)(q - 5)(q - 7)/64 for p ≠ 3, (q - 3)(q - 5)(q - 7)/(3.64) for p = 3. This is a new infinite family of 3-designs.