On the fast computation of the weight enumerator polynomial and the $t$ value of digital nets over finite abelian groups

TL;DR

This paper introduces digital nets over finite abelian groups, providing efficient algorithms for computing their weight enumerator polynomial and strict t-value, extending previous methods over finite fields.

Contribution

It generalizes digital nets to finite abelian groups and develops a MacWilliams type identity for them, enabling efficient computation of t-values and weight enumerator polynomials.

Findings

Computational complexity for t-value: O(N s log N)

Computational complexity for weight enumerator: O(N s (log N)^2)

Precomputing reduces operations for weight enumerator

Abstract

In this paper we introduce digital nets over finite abelian groups which contain digital nets over finite fields and certain rings as a special case. We prove a MacWilliams type identity for such digital nets. This identity can be used to compute the strict $t$-value of a digital net over finite abelian groups. If the digital net has $N$ points in the $s$ dimensional unit cube $[0,1]^s$, then the $t$-value can be computed in $\mathcal{O}(N s \log N)$ operations and the weight enumerator polynomial can be computed in $\mathcal{O}(N s (\log N)^2)$ operations, where operations mean arithmetic of integers. By precomputing some values the number of operations of computing the weight enumerator polynomial can be reduced further.