Decimation classes of nonnegative integer vectors using multisets

TL;DR

The paper generalizes decimation-class counting to nonnegative integer vectors indexed by a finite abelian group $G$ with density $\delta$ under $\gcd(\delta,\ell^{*})=1$, by translating vectors into multisets and extending multiplier theory to multisets. It develops an adjacency-matrix criterion for translate-fixing, analyzes necklaces/bracelets, and introduces $H$-orbits to manage multiplier-group symmetries. A constructive algorithm based on an ILP reformulation and a recursion for the unique-sum problem enables counting decimation classes through a decomposition over multiplier subgroups, aided by a lattice of subgroups of $\mathbb{Z}_{\ell^{*}}^{\times}$. The framework yields practical counts of decimation classes for various $G$ and densities, and extends prior binary-vector results to the multisets setting with rigorous group-theoretic machinery.

Abstract

We describe how previously known methods for determining the number of decimation classes of density $δ$ 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 $δ$ 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.

Theorems & Definitions (26)

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• Theorem 1
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• Definition 1
• Theorem 3