Exploration methods

Various exploration methods are supported by LatNet Builder.

Exploration methods describe how the point sets search space is explored. LatNet Builder supports the following exploration methods:

• evaluation:
  Only the specified point set is evaluated.
• exhaustive:
  All the possible point sets are explored.
• random:
  A random sample of point sets are explored.
• full-CBC:
  Point sets are constructed component-by-component (CBC) in a greedy procedure. Starting from a point set with zero coordinates, coordinates are added one at a time. For each coordinate, all possible coordinate values are evaluated and the best choice is selected to increase the dimension.
• random-CBC:
  Point sets are constructed component-by-component (CBC) in a greedy procedure. Starting from a point set with zero coordinates, coordinates are added one at a time. For each coordinate, a random sample of coordinate values are evaluated and the best choice is selected to increase the dimension.

Additionally, some exploration methods are only supported either for lattices or for digital nets:

• lattices:
  • extend:
    Extend the size of a rank-1 lattice up to a specific modulus.
  • fast-CBC:
    This exploration method can be used to fasten computation for the CBC exploration method when used with the following coordinate-uniform figures of merit:
    • \(\mathcal P_{\alpha}\), \(\mathcal R_\alpha\) and \(\mathcal R\) discrepancies with an \(\ell_2\) norm; and
    • \(B_{\alpha, d, (1)}\) and \(B_{d, (2)}\) interlaced discrepancies with an \(\ell_1\) norm.
      This method [18] uses a Fast Fourier Transform to compute the merit values for all the possible values of the new component of the generating vector.
  • Korobov:
    Explore the generating vectors of the form \( (1, a, a^2, \dots, a^{s-1}) \mod N \) for all \(a\) coprime with \(N\) where \(s\) is the dimension of the lattice and \( N \) its modulus.
  • random Korobov:
    Same as Korobov except that only a random sample of integers/polynomials coprime with \( N \) is explored.
    
    
• digital nets:
  • mixed-CBC:
    This algorithm mixes the full-CBC and the random-CBC algorithms. For the first coordinates, the full-CBC exploration is used and then, it is replaced with the random-CBC exploration. In other words, component-by-component, all the possible coordinate values are tested for the first dimensions but then only a random sample of fixed size is explored for the remaining coordinates.