Affine-invariant strictly cyclic Steiner quadruple systems

A Steiner quadruple system of order v, denoted by SQS(v), is a pair (V, B), where V is a finite set of v points, and B is a collection of 4-subsets of V , called blocks or quadruples, such that each 3-subset (triple) of V is contained in exactly one block in B. An automorphism group of SQS(v) is a permutation group on V leaving B invariant. An SQS(v) is said to be cyclic if it admits an automorphism consisting of a cycle of length v. And then, if the stabilizer of any block is trivial, the system is said to be strictly cyclic, denoted by sSQS(v). It is known that an sSQS(v) exists only if v ≡ 2, 10 (mod 24). But the sufficient existence condition of sSQS(v) is less known so far, all the known results are based on computer search and recursive constructions. Furthermore, let V = Z v , the residue ring modulo v, an sSQS admitting all the units of Z v as multipliers is called affine-invariant. The affine-invariant sSQS is only studied by Köhler (1979) in order to simplify the existence problem of sSQS. Köhler (1982) also considered constructions of affine-invariant 3-(p, 4, λ) designs, for prime p and putative λ > 1. Then, Brand (1993) generalized the constructions to these affine-invariant 3-designs with order of prime powers. All the previously mentioned constructions are depending on the existence of 1-factors of some graphs. In this talk, we denote V = Z 2p ∼ = Z p × Z 2 , where p ≡ 1, 5 (mod 12) is an odd prime. Thus, blocks can be classified into three types. By revisiting some properties of cross-ratio classes of projective geometries, we introduce a graph regarding the cross-ratio classes as vertices, and proves that the graph has a 1-factor, which implies the proper base blocks of the orbits can be obtained. To sum up, we present that, for any prime p ≡ 1, 5 (mod 12), an affine-invariant sSQS(2p) exists. By showing that the recursive construction for a specific class of sSQS by Feng et.al(2008) can be applied also in the case of our affine-invariant sSQS, we prove that, for any odd integer m, if each prime divisor p of m satisfies p ≡ 1, 5 (mod 12), then an sSQS(2m) exists.