On classifying Minkowskian sublattices

TL;DR

This paper extends the classification of quotient codes of Minkowskian sublattices in Euclidean lattices up to dimension 9, providing a deeper understanding of their algebraic and geometric structure.

Contribution

It generalizes the classification of $ ext{Z}/d ext{Z}$-codes of lattice quotients to higher dimensions, specifically up to dimension 9.

Findings

Classification of possible $ ext{Z}/d ext{Z}$-codes in dimension 9

Extension of previous classifications to higher dimensions

Insights into the structure of Minkowskian sublattices

Abstract

Let $\Lambda$ be a lattice in an $n$-dimensional Euclidean space $E$ and let $\Lambda'$ be a Minkowskian sublattice of $\Lambda$, that is, a sublattice having a basis made of representatives for the Minkowski successive minima of $\Lambda$. We extend the classification of possible $\Z/d\Z$-codes of the quotients $\Lambda/\Lambda'$ to dimension~$9$, where $d\Z$ is the annihilator of $\Lambda/\Lambda'$.