Euclidean lattices: theory and applications

TL;DR

This article surveys the theory and applications of Euclidean lattices, emphasizing core invariants such as $\det(L)$, $\delta(L)$, $\Theta(L)$, and $\lambda_i(L)$ that organize both theory and practical uses. It surveys structural topics like well-rounded, eutactic, and perfect lattices, ideal lattices, and counting problems (e.g., Ehrhart theory and integer geometry), while connecting these to coding theory and post-quantum cryptography. The contributions include a synthesis of progress on extreme lattice concepts, WR ideal lattices in cyclic number fields, and concrete lattice-code constructions (Construction A, D, D') as well as lattice-based cryptographic frameworks and attacks. The work underscores the practical impact of lattice theory across dense packing, universal quadratic forms, and secure communications, and presents several open questions, including WR ideal-lattice existence and distribution-reserving reductions in PLWE-type settings.

Abstract

In this editorial survey we introduce the special issue of the journal Communications in Mathematics on the topic in the title of the article. Our main goal is to briefly outline some of the main aspects of this important area at the intersection of theory and applications, providing the context for the articles showcased in this special issue.