3-Designs from PSL(2,q) with cyclic starter blocks

We consider when the projective special linear group over a finite field defines a $3$-design with a cyclic starter block. We will show that the equivalences of the existence of such $3$-$(q+1,5,3)$ and $3$-$(q+1,10,18)$ designs for a prime power $q\equiv 1\pmod{20}$, and $3$-$(q+1,13,33)$ and $3$-$(q+1,26,150)$ designs for a prime power $q\equiv 1\pmod{52}$, respectively.

💡 Research Summary

The paper investigates the existence of block‑transitive 3‑designs arising from the action of the projective special linear group PSL(2,q) on the projective line PG(1,q)=F_q∪{∞}. The authors focus on “starter blocks” that are multiplicative subgroups of the finite field, i.e., sets B={1,β,β²,…,β^{k−1}} where β is a primitive k‑th root of unity and k divides q−1.

When q≡3 (mod 4) the group PSL(2,q) is 3‑transitive, so any k‑subset yields a 3‑design. The interesting case is q≡1 (mod 4), where the action splits the set of 3‑subsets into two orbits O⁺ and O⁻. A necessary and sufficient condition for the PSL(2,q)‑orbit of B to form a 3‑design is that the sum of the quadratic‑character values Δ({x,y,z})=χ((x−y)(y−z)(z−x)) over all triples in B is zero; equivalently the numbers of triples with Δ=+1 and Δ=−1 must be equal in O⁺ and O⁻. Lemma 2.1–2.3 establish that Δ is invariant under PSL(2,q) and that the two orbits are distinguished precisely by the sign of Δ.

The stabilizer of B in PSL(2,q) is shown to be a dihedral group D_{2k}. The authors classify the D_{2k}‑orbits on B³ (Lemma 2.7) into three types and express Δ for each type in terms of the quadratic character of expressions 1−β^m. From this analysis they derive Theorem 2.5: if (q−1)/k is odd then λ=½(k−1)(k−2); if (q−1)/k is even then λ=¼(k−1)(k−2).

A crucial structural result (Theorem 3.1) limits possible k to residues 1,2,5,10,13,17 modulo 24. Computer searches confirm that the only pairs (k,2k) that simultaneously give designs are (5,10) and (13,26). The paper then proves two families of equivalences:

1. Theorem 4.1 (q≡1 (mod 20)).
   The following statements are equivalent:
   (i) (q,5) yields a 3‑design (parameters 3‑(q+1,5,3)).
   (ii) (q,10) yields a 3‑design (parameters 3‑(q+1,10,18)).
   (iii) χ(1+β)=−1, where β=α^{(q−1)/5} and α is a primitive element of F_q.
   (iv) There exists θ∈F_q^× with χ(θ)=−1 satisfying θ²−4θ−1=0.
   (v) 5∉⟨α⁴⟩ (i.e., 5 is not a fourth power in F_q).
   (vi) The prime p (if q=p) cannot be represented as p=x²+20y².
   (vii) The prime p cannot be represented as p=x²+100y².

   The proof computes Δ for the two D_{10}‑orbits on B³ when k=5 and shows that the total sum vanishes exactly when χ(1+β)=−1. For k=10 the same condition appears after a more elaborate analysis of eight D_{20}‑orbits. The equivalence between (iii) and (iv) uses the explicit solutions θ₀,θ₁ of the quadratic equation in terms of β, while (iii)↔(v) follows from the identity (β(1−β)²(1+β))²=5. Classical number‑theoretic results (Brink, Hasse) translate the character condition into the non‑representability statements (vi) and (vii).

2. Theorem 5.1 (q≡1 (mod 52)).
   Analogously, (q,13) yields a 3‑design (parameters 3‑(q+1,13,33)) if and only if (q,26) yields a 3‑design (parameters 3‑(q+1,26,150)). The equivalent character condition is χ(1+γ)=−1 where γ=α^{(q−1)/13}. The proof mirrors the k=5,10 case, using the D_{26}‑action on B³ and checking that the Δ‑sum vanishes precisely under the same character condition.

The paper also studies field extensions. Proposition 3.3 shows that if (q,k) gives a 3‑design, then for any odd integer n, (qⁿ,k) also gives a 3‑design, because the primitive k‑th root and the quadratic character behave compatibly under odd extensions. Conversely, Proposition 3.5 proves that for even n no such designs exist, as the quadratic character becomes trivial on the base field, forcing Δ≡+1 for all triples.

In summary, the authors provide a unified framework that connects previously isolated examples (Li‑Deng‑Zhang’s 3‑(q+1,5,3) designs and Bonnecaze‑Solé’s 3‑(q+1,10,18) designs) through a common character condition. They extend the theory to the (13,26) pair, give precise number‑theoretic criteria for the existence of these designs, and clarify how the property behaves under field extensions. The work deepens the understanding of how PSL(2,q) can generate block‑transitive 3‑designs with cyclic starter blocks and opens avenues for further exploration of higher‑order designs or other permutation groups.