Related papers: Modular Lattices from a Variation of Construction …
We focus on two important classes of lattices, the well-rounded and the cyclic. We show that every well-rounded lattice in the plane is similar to a cyclic lattice, and use this cyclic parameterization to count planar well-rounded…
Motivated by sparseness conditions for holographic CFTs, we investigate sparseness of modular forms, lattices, and codes. For this we investigate the free energy of such objects as their weight, dimension or size goes to infinity. We…
This is a survey of characterizations and relationships between some properties of lattices, particularly the modular, Arguesian, linear, and distributive properties, but also some other related properties. The survey emphasizes finite and…
In this paper, we present a lattice-theoretic characterization for valuated matroids, which is an extension of the well-known cryptomorphic equivalence between matroids and geometric lattices ($=$ atomistic semimodular lattices). We…
Conference matrices are used to define complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by non-principal ideals yields simple…
In an earlier paper (math.NT/9906019) we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n has a characteristic vector of norm at most n-8. [A "characteristic vector" of L is a vector w in L such that…
In this paper, using compute-and-forward as an example, we provide an overview of constructions of lattices from codes that possess the right algebraic structures for harnessing interference. This includes Construction A, Construction D,…
We study the shortest vector lengths in module lattices over arbitrary number fields, with an emphasis on cyclotomic fields. In particular, we sharpen the techniques of arXiv:2308.15275v2 to establish improved results for the variance of…
We study ideal lattices constructed from totally definite quaternion algebras over totally real number fields, and generalize the definition of Arakelov-modular lattices over number fields. In particular, we prove for the case where the…
A Lattice is a partially ordered set where both least upper bound and greatest lower bound of any pair of elements are unique and exist within the set. K\"{o}tter and Kschischang proved that codes in the linear lattice can be used for error…
This paper partially addresses the problem of characterizing the lengths of vectors in a family of Euclidean lattices that arise from any CM number field. We define a modified quadratic form on these lattices, the weighted norm, that…
Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. The design of lattices for BF channels offers a challenging problem, which differs greatly from its…
Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices. In this work, we study such lattices and their reduction algorithms. Firstly, when the lattice is spanned over a two…
Periodic lattices are natural generalizations of lattices, which arise naturally in diophantine approximations with rationals of bounded denominators. In this paper, we prove analogues of classical theorems in geometry of numbers for…
Characteristic Lie rings for Toda type 2+1 dimensional lattices are defined. Some properties of these rings are studied. Infinite sequence of special kind modules are introduced. It is proved that for known integrable lattices these modules…
We propose a coding scheme that achieves the capacity of the compound MIMO channel with algebraic lattices. Our lattice construction exploits the multiplicative structure of number fields and their group of units to absorb ill-conditioned…
This article is the second part of an essay dedicated to lattices freely generated by posets within a variety. The first part dealt with four easy varieties while this part is concerned with finitely generated varieties. Here we present a…
Part B (of a project involving four Parts) is about "bases of lines", a concept introduced by C. Herrmann and the author in the late 80's. Bases of lines attempt to describe a given modular lattice in a geometric way akin to how projective…
We construct multilevel lattice codes from multiquadratic number fields for the compound block-fading wiretap channel. More precisely, we specialize Construction $\pi_A$ over the ring of integers $\mathcal{O}_K$ and exploit rational primes…
We give a new proof of the fact that any finite quadratic module can be decomposed into indecomposable ones. For any indecomposable finite quadratic module, we construct a lattice, and a positive definite lattice, both of which are of the…