On the fast computation of the weight enumerator polynomial and the \(t\) value of digital nets over finite abelian groups (Q2870510)

The authors give a generalization of the concept of digital nets by introducing \((t,m,s)-\)nets over finite abelian groups. In this setting they also develop a duality theory and show how it can be applied to obtain the quality parameter \(t\) of a digital net. This is related to the computation of the weight enumerator polynomial of the dual of a given digital net. Indeed, in the case when the digital net has \(N\) points in \([0,1)^s\) the authors provide two algorithms that allow to compute the weight enumerator polynomial and the \(t-\)parameter in \(\mathcal{O}(Ns(\log N)^2)\) and \(\mathcal{O}(Ns\log N)\) operations, respectively. Furthermore, the last section of this paper contains a generalization of the notion of dual net to the case when the digital net is defined over a finite ring. The authors also determine for which finite rings the ring-theoretic dual net coincides with the character-theoretic dual net used for digital nets over finite abelian groups.