The Hermite normal form for certain rank-1 circulant and skew-circulant lattice rules

Characterisations are provided of the Hermite normal form of certain integer circulant and skew-circulant matrices. These matrices are associated with rank-1 circulant and skew-circulant lattice rules. Previous computer searches for lattice rules of specified trigonometric degree have indicated that there is some merit in searches of such lattice rules.We also consider the question of whether a Hermite normal form with the characterisation for these rank-1 circulant lattice rules is actually associated to a circulant lattice rule. Though the answer is negative in the general case, several examples where the answer is positive will be given.