Wall Lattices Research Articles

Algebraic constructions of rotated unimodular lattices and direct sum of Barnes–Wall lattices

In this paper, we construct some families of rotated unimodular lattices and rotated direct sum of Barnes–Wall lattices [Formula: see text] for [Formula: see text] and [Formula: see text] via ideals of the ring of the integers [Formula: see text] for [Formula: see text] and [Formula: see text]. We also construct rotated [Formula: see text] and [Formula: see text]-lattices via [Formula: see text]-submodules of [Formula: see text]. Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. The minimum product distances of such constructions are also presented here.

Strongly perfect lattices sandwiched between Barnes–Wall lattices

New series of $2^{2m}$-dimensional universally strongly perfect lattices $\Lambda_I $ and $\Gamma_J $ are constructed with $$2BW_{2m} ^{\#} \subseteq \Gamma _J \subseteq BW_{2m} \subseteq \Lambda _I \subseteq BW _{2m}^{\#} .$$ The lattices are found by restricting the spin representations of the automorphism group of the Barnes-Wall lattice to its subgroup ${\mathcal U}_m:={\mathcal C}_m (4^H_{\bf 1}) $. The group ${\mathcal U}_m$ is the Clifford-Weil group associated to the Hermitian self-dual codes over ${\bf F} _4$ containing ${\bf 1}$, so the ring of polynomial invariants of ${\mathcal U}_m$ is spanned by the genus-$m$ complete weight enumerators of such codes. This allows us to show that all the ${\mathcal U}_m$ invariant lattices are universally strongly perfect. We introduce a new construction, $D^{(cyc)}$ for chains of (extended) cyclic codes to obtain (bounds on) the minimum of the new lattices.

TRELLIS-CODED MODULATION FOR LINEAR DISTORTION CHANNELS

Introduction: Real communication channels used for high-speed data transmission are characterized by substantial linear distortions leading to a significant decrease in the effectiveness of the known coded modulation techniques developed for distortion-free channels. This means that a modified coded modulation technique should be created, which would take into account the channel distortions. Purpose: Using multidimensional signal sets to build up multilevel trellis-coded modulation constructions with multidimensional constellations and multi-stage decoding, matched to the channels with linear distortions, and study of their main characteristics. Results: A new approach was developed to the design of trellis-coded modulation constructions for channels with linear distortions, based on the combination of two techniques: decomposition of the channel into a set of independent memoryless subchannels and time-frequency multilevel coding. The high level of flexibility of the proposed trellis-coded modulation constructions makes it possible to effectively take into account linear channel distortions and compensate for them, due to the rational choice of the number of the subchannels and optimization of the energy distribution between the subchannels. It also provides a wide-range tradeoff between performance and decoding complexity. The relationships between the basic parameters of the proposed class of trellis-coded modulation constructions were obtained in an explicit form. These relationships allow you to provide tradeoffs between the transmission rate, the minimum distance, and the decoding complexity. Examples of multilevel trellis-coded modulation constructions based on multidimensional Barnes — Wall lattices are given. A comparative study showed that as the value of the permissible decoding complexity increases, constructions with larger values of both the number of subchannels and the cardinality of the signal constellation become more preferable. Practical significance: The proposed class of trellis-coded modulation constructions allows you to effectively take into account and to overcome the influence of linear channel distortions common in communication channels used for high-speed data transmission. An additional advantage of these constructions is a relatively small value of the crest factor of transmitted signals and thereby more efficient use of the transmitter's power.

Extra-Dimensional “Metamaterials”: A Model of Inflation Due to a Metric Signature Transition

Lattices of topological defects, such as Abrikosov lattices and domain wall lattices often arise as metastable ground states in higher-dimensional field theoretical models. We demonstrate that such lattice states may be described as extra-dimensional metamaterials via higher-dimensional effective medium theory. A 4+1 dimensional extension of Maxwell electrodynamics with a compactified time-like dimension has been considered as an example. It is demonstrated that from the point of view of macroscopic electrodynamics an Abrikosov lattice state in such a 4+1 dimensional spacetime may be described as a uniaxial hyperbolic metamaterial. Extraordinary photons perceive this medium as a 3+1 dimensional Minkowski spacetime in which one of the original spatial dimensions (the optical axis of the metamaterial) plays the role of a new time-like coordinate. Since metric signature of this effective space-time depends on the Abrikosov lattice periodicity, the described model may be useful in studying metric signature transitions. A particular kind of metric signature transition understood as a macroscopic medium effect may emulate cosmological inflation.

List-Decoding Barnes–Wall Lattices

The question of list-decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete linear structure of linear codes and point lattices in $${\mathbb{R}^{N}}$$RN, and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the study of list decoding for lattices. Namely: for a lattice $${\mathcal{L}\subseteq \mathbb{R}^N}$$L⊆RN, given a target vector $${r \in \mathbb{R}^N}$$rźRN and a distance parameter d, output the set of all lattice points $${w \in \mathcal{L}}$$wźL that are within distance d of r. In this work, we focus on combinatorial and algorithmic questions related to list decoding for the well-studied family of Barnes---Wall lattices. Our main contributions are twofold: 1.We give tight combinatorial bounds on the worst-case list size, showing it to be polynomial in the lattice dimension for any error radius bounded away from the lattice's minimum distance (in the Euclidean norm).2. We use our combinatorial bounds to generalize the unique-decoding algorithm of Micciancio and Nicolosi (IEEE International Symposium on Information Theory 2008) to work beyond the unique-decoding radius and still run in polynomial time up to the list-decoding radius. Just like the Micciancio---Nicolosi algorithm, our algorithm is highly parallelizable, and with sufficiently many processors, it can run in parallel time only poly-logarithmic in the lattice dimension.

Harnessing diffraction grating in an in-plane switching cell submitted to zigzag lattice.

Programmable diffraction gratings are relevant in optical data processing. One of the adequate device candidates is the in-plane switching liquid crystal cell. This technology, developed initially for liquid crystal screens, has also been studied with two inter-digital electrodes as a diffraction grating. Recently, the apparition of programmable zigzag wall lattices in an in-plane switching configuration has been reported. Here, we report a theoretical and experimental study of programmable diffraction grating in an in-plane switching cell.

A NEW LOWER BOUND FOR HERMITE'S CONSTANT FOR SYMPLECTIC LATTICES

In this paper we give an improved lower bound on Hermite's constant δ2g for symplectic lattices in even dimensions (g = 2n) by applying a mean-value argument from the geometry of numbers to a subset of symmetric lattices. We obtain only a slight improvement. However, we believe that the method applied has further potential. Furthermore, in this paper we present new families of highly symmetric (symplectic) lattices, which occur in dimensions of powers of two. The lattices in dimension 2n are constructed with the help of a multiplicative matrix group isomorphic to (ℤ2n, +). We furthermore show the connection of these lattices with the circulant matrices and the Barnes–Wall lattices.

Nucleon electromagnetic form factors from lattice QCD using2+1flavor domain wall fermions on fine lattices and chiral perturbation theory

We present a high-statistics calculation of nucleon electromagnetic form factors in $N_f=2+1$ lattice QCD using domain wall quarks on fine lattices, to attain a new level of precision in systematic and statistical errors. Our calculations use $32^3 \times 64$ lattices with lattice spacing a=0.084 fm for pion masses of 297, 355, and 403 MeV, and we perform an overdetermined analysis using on the order of 3600 to 7000 measurements to calculate nucleon electric and magnetic form factors up to $Q^2 \approx$ 1.05 GeV$^2$. Results are shown to be consistent with those obtained using valence domain wall quarks with improved staggered sea quarks, and using coarse domain wall lattices. We determine the isovector Dirac radius $r_1^v$, Pauli radius $r_2^v$ and anomalous magnetic moment $\kappa_v$. We also determine connected contributions to the corresponding isoscalar observables. We extrapolate these observables to the physical pion mass using two different formulations of two-flavor chiral effective field theory at one loop: the heavy baryon Small Scale Expansion (SSE) and covariant baryon chiral perturbation theory. The isovector results and the connected contributions to the isoscalar results are compared with experiment, and the need for calculations at smaller pion masses is discussed.

Midwest cousins of Barnes–Wall lattices

Given a rational lattice and suitable set of linear transformations, we construct a cousin lattice. Sufficient conditions are given for integrality, evenness and unimodularity. When the input is a Barnes–Wall lattice, we get multi-parameter series of cousins. There is a subseries consisting of unimodular lattices which have ranks 2 d − 1 ± 2 d − k − 1 , for odd integers d ⩾ 3 and integers k = 1 , 2 , … , d − 1 2 . Their minimum norms are moderately high: 2 ⌊ d 2 ⌋ − 1 .

Extreme lattices and vexillar designs

We define a notion of vexillar design for the flag variety in the spirit of the already known spherical designs. We explain how the orbits of any flag under the action of a finite group can be a design. We show that a lattice is locally optimal for the general Hermite constant when its minima form a 4-design. The reasoning proves useful to show the extremality of many new expected examples ( E 8 , Λ 24 , Barnes–Wall lattices, Thompson–Smith lattice for instance) that were out of reach until now.

Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

H. Cohn et al. proposed an association scheme of 64 points in R 14 which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal collections of real mutually unbiased bases. These schemes are also related to extremal line-sets in Euclidean spaces and Barnes–Wall lattices. D. de Caen and E.R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.

Stability of winding cosmic wall lattices with X type junctions

This work confirms the stability of a class of domain wall lattice models that can produce accelerated cosmological expansion, with pressure to density ratio w = −1/3 at early times, and with w = −2/3 at late times when the lattice scale becomes large compared to the wall thickness. For walls of tension TI, the relevant X type junctions could be unstable (for a sufficiently acute intersection angle α) against separation into a pair of Y type junctions joined by a compound wall, only if the tension of the latter were less than 2TI (and for an approximately right-angled intersection if it were less than ) which cannot occur in the class considered here. In an extensive category of multicomponent scalar field models of forced harmonic (linear or nonlinear) type it is shown how the relevant tension—which is the same as the surface energy density U of the wall—can be calculated as the minimum (geodesic) distance between the relevant vacuum states as measured on the space of field values Φi using a positive definite (Riemannian) energy metric that is obtained from the usual kinetic metric (which is flat for a model with an ordinary linear kinetic part) by application of a conformal factor proportional to the relevant potential function V. For suitably periodic potential functions there will be corresponding periodic configurations—with parallel walls characterized by incrementation of a winding number—in which the condition for stability of large scale bunching modes is shown to be satisfied automatically. It is suggested that such a configuration—with a lattice lengthscale comparable to intergalactic separation distances—might have been produced by a late stage of cosmological inflation.

Few-cosine spherical codes and Barnes–Wall lattices

Using Barnes–Wall lattices and 1-cocycles on finite groups of monomial matrices, we give a procedure to construct tricosine spherical codes. This was inspired by a 14-dimensional code which Ballinger, Cohn, Giansiracusa and Morris discovered in studies of the universally optimal property. Their code has 64 vectors and cosines − 3 7 , − 1 7 , 1 7 . We construct the Optimism Code, a 4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are 0 , ± 1 4 , − 1 . Its automorphism group has shape 2 1 + 8 ⋅ GL ( 4 , 2 ) . The Optimism Code contains a subcode related to the BCGM code. The Optimism Code implies existence of a nonlinear binary code with parameters ( 16 , 256 , 6 ) , a Nordstrom–Robinson code, and gives a context for determining its automorphism group, which has form 2 4 : Alt 7 .

Nanoscopic Liquid Bridges between Chemically Patterned Atomistic Walls

A binary liquid mixture, containing the Lennard-Jones molecules A and B, in equilibrium with a bulk liquid reservoir near the point of phase separation, confined between atomistic chemically patterned walls, is studied by grand canonical Monte Carlo simulations. In the bulk, the B-rich phase is stable and the A-rich phase is metastable. The walls bear patches attractive to A; when the walls are close, A-rich liquid bridges condense between the patches. The normal and lateral forces on the walls are measured as a function of the wall separation and of the lateral displacement between the patches on opposite walls. When there are one or two molecular layers in the bridge and the wall lattice constant is close to that of crystalline A, the normal and lateral forces depend strongly on the registry of the wall lattices, varying in an oscillatory manner.

Augmented Product Codes and Lattices: Reed–Muller Codes and Barnes–Wall Lattices

This paper concerns the construction of the so-called augmented product codes and augmented product lattices. These are obtained by augmenting product codes or product lattices from certain classes thus obtaining higher dimensional codes or lattices from the same class, respectively. Certain properties of the augmented product construction are derived, and specific construction examples are given. In particular, it is shown that the Reed-Muller codes, the Golay code, the Barnes-Wall lattices, as well as the Leech lattice all have various augmented product constructions.

Pieces of [formula omitted]: existence and uniqueness for Barnes–Wall and Ypsilanti lattices

We give a new existence proof for the rank 2 d even lattices usually called the Barnes–Wall lattices, and establish new results on uniqueness, structure and transitivity of the automorphism group on certain kinds of sublattices. Our proofs are relatively free of calculations, matrix work and counting, due to the uniqueness viewpoint. We deduce the labeling of coordinates on which earlier constructions depend. Extending these ideas, we construct in dimensions 2 d , for d ≫ 0 , the Ypsilanti lattices, which are families of indecomposable even unimodular lattices which resemble the Barnes–Wall lattices. The number ϒ ( 2 d ) of isometry types here is large: log 2 ( ϒ ( 2 d ) ) has dominant term at least r 4 d 2 2 d , for any r ∈ [ 0 , 1 2 ) . Our lattices may be the first explicitly given families whose sizes are asymptotically comparable to the Siegel mass formula estimate ( log 2 ( mass ( n ) ) has dominant term 1 4 log 2 ( n ) n 2 ). This work continues our general uniqueness program for lattices, begun in Pieces of Eight (Adv. in Math. 148 (1999) 75). See also our new uniqueness proof for the E 8 -lattice (J. Number Theory 103 (2003) 77).

Spontaneous formation of domain wall lattices in two spatial dimensions

We show that the process of spontaneous symmetry breaking can trap a field theoretic system in a highly non-trivial state containing a lattice of domain walls. In one large compact space dimension, a lattice is inevitably formed. In two dimensions, the probability of lattice formation depends on the ratio of sizes L_x, L_y of the spatial dimensions. We find that a lattice can form even if R=L_y/L_x is of order unity. We numerically determine the number of walls in the lattice as a function of L_x and L_y.

Formation of domain wall lattices

We study the formation of domain walls in a phase transition in which an ${S}_{5}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry is spontaneously broken to ${S}_{3}\ifmmode\times\else\texttimes\fi{}{S}_{2}.$ In one compact spatial dimension we observe the formation of a stable domain wall lattice. In two spatial dimensions we find that the walls form a network with junctions, there being six walls to every junction. The network of domain walls evolves so that junctions annihilate antijunctions. The final state of the evolution depends on the relative dimensions of the simulation domain. In particular we never observe the formation of a stable lattice of domain walls for the case of a square domain but we do observe a lattice if one dimension is somewhat smaller than the other. During the evolution, the total wall length in the network decays with time as ${t}^{\ensuremath{-}0.71},$ as opposed to the usual ${t}^{\ensuremath{-}1}$ scaling typical of regular ${Z}_{2}$ networks.

The Commensurate-Incommensurate Transition of Deuterium Monolayers on Graphite

Specific heat and neutron diffraction measurements have been made to study in detail the commensurate-incommensurate (C-IC) transition of ortho-deuterium monolayers physisorbed on graphite. Several novel intermediate phases between the commensurate and the incommensurate phase have been observed, which we denote as α, β, γ, δ and ε phases. The neutron diffraction results reveal that these phases can be described by domain wall lattices of different types (striped or hexagonal), and thus provide an experimental verification of recent C-IC transition theories.

Disclination Structures in Bloch Wall Lattices in BaFe12O19 and SmCo5

Domain walls in barium hexaferrite (BaFe12O19) and SmCo5 are shown to form a lattice. Defects observed in the Boch wall patterns are interpreted geometrically as dislocations and disclinations. The elastic properties of these defects are briefly discussed. The faceted appearance of some Bloch wall patterns is analyzed on the basis of a surface energy torque term arising from an anisotropy in the Bloch wall energy. The anisotropy is deduced to arise from high-order coefficients in the anisotropy energy and the magnetostriction.