Barnes-Wall Lattices Research Articles

The Voronoi Region of the Barnes–Wall Lattice Λ16

The Voronoi Region of the Barnes–Wall Lattice Λ₁₆

On the Decoding of Lattices Constructed via a Single Parity Check

This paper investigates the decoding of a remarkable set of lattices: We treat in a unified framework the Leech lattice in dimension 24, the Nebe lattice in dimension 72, and the Barnes-Wall lattices. A new interesting lattice, named <inline-formula> <tex-math notation="LaTeX">$L_{3\cdot 24}$ </tex-math></inline-formula>, is constructed as a simple application of the single parity check on the Leech lattice. The common aspect of these lattices is that they can be obtained via a single parity check or via the <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-ing construction. We exploit these constructions to introduce a new efficient paradigm for decoding. This leads to efficient list decoders and quasi-optimal decoders on the Gaussian channel. Both theoretical and practical performance (point error probability and complexity) of the new decoders are provided.

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.

Low dimensional strongly perfect lattices IV: The dual strongly perfect lattices of dimension 16

We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice Λ16, the extremal 5-modular lattice N16, the odd Barnes-Wall lattice O16 and its dual, and one pair of new lattices Γ16 and its dual. The latter pair belongs to a new infinite series of dual strongly perfect lattices, the sandwiched Barnes-Wall lattices, described by the authors in a previous paper. An updated table of all known strongly perfect lattices up to dimension 26 is available in the catalogue of lattices [15].

Concatenated Coding for Multilevel Flash Memory with Low Error Correction Capabilities in Outer Stage

One of the approaches to organization of error correcting coding for multilevel flash memory is based on concatenated construction, in particular, on multidimensional lattices for inner coding. A characteristic feature of such structures is the dominance of the complexity of the outer decoder in the total decoder complexity. Therefore the concatenated construction with low-complexity outer decoder may be attractive since in practical applications the decoder complexity is the crucial limitation for the usage of the error correction coding.&#x0D; We consider a concatenated coding scheme for multilevel flash memory with the Barnes-Wall lattice based codes as an inner code and the Reed-Solomon code with correction up to 4…5 errors as an outer one.&#x0D; Performance analysis is fulfilled for a model characterizing the basic physical features of a flash memory cell with non-uniform target voltage levels and noise variance dependent on the recorded value (input-dependent additive Gaussian noise, ID-AGN). For this model we develop a modification of our approach for evaluation the error probability for the inner code. This modification uses the parallel structure of the inner code trellis which significantly reduces the computational complexity of the performance estimation. We present numerical examples of achievable recording density for the Reed-Solomon codes with correction up to four errors as the outer code for wide range of the retention time and number of write/read cycles.

Lattices Associated with Octonion Algebras

In this paper, we propose a construction of lattices in&nbsp;dimension 8n via octonion orders over a totally real number field.&nbsp;To show the potential of this construction, we present rotated&nbsp;versions of $E_8$,&nbsp;$\Lambda_{16}$, the&nbsp; densest known lattices in these dimensions&nbsp;and a lattice with the same density of the Barnes-Wall lattice in&nbsp;dimension 32. Perspectives applications of these lattices based on&nbsp;octonion algebras are in lattice-based cryptography and lattice&nbsp;coding for MIMO and MISO channels.

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.

Dense packings from algebraic number fields and codes

We introduce a new method from number fields and codes to construct dense packings in the Euclidean spaces. Via the canonical Q-embedding of arbitrary number field K into R[K:Q], both the prime ideal p and its residue field κ can be embedded as discrete subsets in R[K:Q]. Thus we can concatenate the embedding image of the Cartesian product of n copies of p together with the image of a length n code over κ. This concatenation leads to a packing in the Euclidean space Rn[K:Q]. Moreover, we extend the single concatenation to multiple concatenations to obtain dense packings and asymptotically good packing families. For instance, with the help of Magma, we construct a 256-dimensional packing denser than the Barnes–Wall lattice BW256.

The automorphism group of the -orbifold of the Barnes–Wall lattice vertex operator algebra of central charge 32

AbstractIn this paper, we prove that the full automorphism group of the${\mathbb Z}_2$-orbifold of the Barnes–Wall lattice vertex operator algebra of central charge 32 has the shape 227.E6(2). In order to identify the group structure, we introduce a graph structure on the Griess algebra and show that it is a rank 3 graph associated toE6(2).

Practical Encoders and Decoders for Euclidean Codes from Barnes-Wall Lattices

We address the application of Barnes-Wall (BW) lattice codes for communication over additive white Gaussian noise (AWGN) channels. We introduce Construction A^prime of complex BW lattices that makes new connection between linear codes over polynomial rings and lattices. We show that Construction A^prime of BW lattices is equivalent to the multilevel construction from Reed-Muller codes proposed by Forney. To decode the BW lattice code, we adapt the low-complexity sequential BW lattice decoder (SBWD) proposed by Micciancio and Nicolosi. First we study the error performance of SBWD for decoding the infinite lattice, and demonstrate that it is powerful in making correct decisions well beyond the packing radius. Subsequently, we use the SBWD to decode lattice codes through a novel noise trimming technique, where the received vector is appropriately scaled before applying the SBWD. We show that the noise trimming technique is most effective for decoding BW lattice codes in smaller dimensions, while the gain diminishes for decoding codes in larger dimensions.

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.

Multipartite entanglement arising from dense Euclidean lattices in dimensions 4–24

The group of automorphisms of Euclidean (embedded in ) dense lattices such as the root lattices D4 and E8, the Barnes–Wall lattice BW16, the unimodular lattice D12+ and the Leech lattice Λ24 may be generated by entangled quantum gates of the corresponding dimension. These (real) gates/lattices are useful for quantum error correction: for instance, the two- and four-qubit real Clifford groups are the automorphism groups of the lattices D4 and BW16, respectively, and the three-qubit real Clifford group is maximal in the Weyl group W(E8). Technically, the automorphism group Aut(Λ) of the lattice Λ is the set of orthogonal matrices B such that, following the conjugation action by the generating matrix of the lattice, the output matrix is unimodular (of determinant ±1, with integer entries). When the degree n is equal to the number of basis elements of Λ, Aut(Λ) also acts on basis vectors and is generated with matrices B such that the sum of squared entries in a row is 1, i.e. B may be seen as a quantum gate. For the dense lattices listed above, maximal multipartite entanglement arises. In particular, one finds a balanced tripartite entanglement in E8 (the two- and three-tangles have the same magnitude 1/4) and a Greenberger–Horne–Zeilinger-type entanglement in BW16. In this paper, we also investigate the entangled gates from D12+ and Λ24, by seeing them as systems coupling a qutrit to two- and three-qubits, respectively. In addition to quantum computing, the work may be related to particle physics in the spirit of Planat et al (2011 Rep. Math. Phys.66 39–51).

Rigidity of spherical codes

A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity of spherical codes, particularly kissing configurations. One surprise is that the kissing configuration of the Coxeter-Todd lattice is not jammed, despite being locally jammed (each individual cap is held in place if its neighbors are fixed); in this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic lattice in three dimensions. By contrast, we find that many other packings have jammed kissing configurations, including the Barnes-Wall lattice and all of the best kissing configurations known in four through twelve dimensions. Jamming seems to become much less common for large kissing configurations in higher dimensions, and in particular it fails for the best kissing configurations known in 25 through 31 dimensions. Motivated by this phenomenon, we find new kissing configurations in these dimensions, which improve on the records set in 1982 by the laminated lattices.

A Mordell inequality for lattices over maximal orders

In this paper we prove an analogue of Mordell's inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic number field or a totally definite rational quaternion algebra. This inequality implies that the 16-dimensional Barnes-Wall lattice has optimal density among all 16-dimensional lattices with Hurwitz structures.

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.

On lattices whose minimal vectors form a 6-design

Let L be a lattice of dimension n ≤ 24 such that the minimal vectors of L form a 6-design and generate L . Then L is similar to either the root lattice E 8 , the Barnes–Wall lattice BW 16 , the Leech lattice Λ 24 , or n = 23 . For n = 23 we conjecture that the only possibilities for L are the shorter Leech lattice O 23 or its even sublattice Λ 23 .

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.

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 .