The number of lattice rules of specified upper class and rank (Q1415408)

The authors study cubature over the \(s\)-dimensional unit cube using lattice rules. There exists a one-to-one correspondence between lattice rules and the upper triangular lattice form \(B\), which is a non-singular, upper triangular, \(s\times s\) integer matrix. One notion to classify lattice rules is the so-called {lattice rank}. Another way to define groups of lattice rules is the {upper class} of a lattice rule. It is the product \[ \nu_s([{\mathbf b}])=b_{22}b_{33}^2\cdots b_{ss}^{s-1} \] of diagonal entries of \(B\) [cf. \textit{J. N. Lyness}, \textit{T. Sørevnik} and \textit{P. Keast}, Math. Comput. 56, No. 193, 243--255 (1991; Zbl 0713.65015)]. The main aim of the paper under review is to find the number of lattice rules of a given rank within a given upper class. For prime upper classes this is done by deriving suitable recurrence relations for the number \(\nu_s([{\mathbf b}])\), while \(p\)-Sylow decomposition allows to extend the result to the general case.