Equivalence classes of matchings and lattice-square designs

Abstract

We enumerate nonisomorphic lattice-square designs yielded by a conventional construction. Constructed designs are specified by words composed from finite-field elements. These words are permuted by the isomorphism group in question. The latter group contains a direct-product subgroup, acting, respectively, upon the positions and identities of the finite-field elements. We review enumeration theory for such direct-product groups. This subgroup is a direct product of a hyperoctahedral and a dihedral group, with the orbits of the hyperoctahedral group, acting on the positions of the field elements, interpretable as perfect matchings. Thus, the enumeration of dihedral equivalence classes of perfect matchings provides an upper bound on the number of nonisomorphic, constructed designs. The full isomorphism group also contains non-direct-product elements, and the isomorphism classes are enumerated using Burnside's Lemma: counting the number of orbits of a normal subgroup fixed by the quotient group. This approach is applied to constructed lattice-square designs of odd, prime-power order ⩽13.