Decimation classes of nonnegative integer vectors using multisets

TL;DR

This paper extends methods for counting decimation classes from binary vectors to nonnegative integer vectors over finite abelian groups, introducing a generalized multiplier theory with applications to group translations.

Contribution

It generalizes the theory of decimation classes and multipliers to multisets in finite abelian groups, providing new tools for analyzing vector classes under group actions.

Findings

Extended decimation class counting to nonnegative integer vectors

Developed a generalized multiplier theory for multisets in finite abelian groups

Provided conditions for group elements to fix translations

Abstract

We describe how previously known methods for determining the number of decimation classes of density $\delta$ binary vectors can be extended to nonnegative integer vectors, where the vectors are indexed by a finite abelian group $G$ of size $\ell$ and exponent $\ell^*$ such that $\delta$ is relatively prime to $\ell^*$. We extend the previously discovered theory of multipliers for arbitrary subsets of finite abelian groups, to arbitrary multisubsets of finite abelian groups. Moreover, this developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing.