Unimodular Lattices Research Articles

Codes over rings and Hermitian lattices

The purpose of this paper is to study a further connection between linear codes over three kinds of finite rings and Hermitian lattices over a complex quadratic field $$K={\mathbb {Q}}(\sqrt{-\ell })$$K=Q(-l), where $$\ell >0$$l>0 is a square free integer such that $$\ell \equiv 3 \pmod {4}.$$l?3(mod4). Shaska et al. (Finite Fields Appl 16(2): 75---87, 2010) consider a ring $${\mathcal {R}}=\mathcal{O}_K / p \mathcal{O}_K$$R=OK/pOK (p is a prime) and study Hermitian lattices constructed from codes over the ring $${\mathcal {R}}$$R. We consider a more general ring $${\mathcal {R}}=\mathcal{O}_K / p^e \mathcal{O}_K$$R=OK/peOK, where $$e \ge 1$$e?1. Using $$p^e$$pe allows us to make a connection from a code to a much larger family of lattices. That is, we are not restricted to those lattices whose minimum norm is less than p. We first show that $${\mathcal {R}}$$R is isomorphic to one of the following three non-isomorphic rings: a Galois ring $$GR(p^e, 2)$$GR(pe,2), $$ {\mathbb {Z}}_{p^e} \times {\mathbb {Z}}_{p^e}$$Zpe×Zpe, and $${\mathbb {Z}}_{p^e} + u {\mathbb {Z}}_{p^e}$$Zpe+uZpe. We then prove that the theta functions of the Hermitian lattices constructed from codes over these three rings are determined by the complete weight enumerators of those codes. We show that self-dual codes over $${\mathcal {R}}$$R produce unimodular Hermitian lattices. We also discuss the existence of Hermitian self-dual codes over $${\mathcal {R}}$$R. Furthermore, we present MacWilliams' relations for codes over $${\mathcal {R}}$$R.

Some Results Related to the Conjecture by Belfiore and Solé

In the first part of this paper, we consider the relation between kissing number and the secrecy gain. We show that on an n=24m+8k -dimensional even unimodular lattice, if the shortest vector length is at least 2m , then as the number of vectors of length 2m decreases, the secrecy gain increases. We will also prove a similar result on general unimodular lattices. We will also consider the situations with shorter vectors. Furthermore, assuming the conjecture by Belfiore and Solé, we will calculate the difference between inverses of secrecy gains as the number of vectors varies. We will show by an example that there exist two lattices in the same dimension with the same shortest vector length and the same kissing number, but different secrecy gains. Finally, we consider some cases of a question by Elkies by providing an answer for a special class of lattices assuming the conjecture by Belfiore and Solé. We will also get a conditional improvement on some Gaulter's results concerning the conjecture.

Extreme value theory for random walks on homogeneous spaces

In this paper we study extreme events for random walks on homogeneous spaces. We consider the following three cases. On the torus we study closest returns of a random walk to a fixed point in the space. For a random walk on the space of unimodular lattices we study extreme values for lengths of the shortest vector in a lattice. For a random walk on a homogeneous space we study the maximal distance a random walk gets away from an arbitrary fixed point in the space. We prove an exact limiting distribution on the torus and upper and lower bounds for sparse subsequences of random walks in the two other cases. In all three settings we obtain a logarithm law.

Arithmetic volumes for lattices over p-adic rings

In this paper we develop formulas for computing the arithmetic volume of orthogonal groups for lattices over the maximal orders of finite extensions of Zp. We specifically develop new explicit formulas for unimodular lattices over 2-adic rings. We also develop a reduction of the general problem to that of unimodular lattices together with the combinatorial problem of computing representatives for all possible Jordan decompositions.

ON AUTOMORPHISMS OF EXTREMAL EVEN UNIMODULAR LATTICES

The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are determined using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order m with φ(m) > 24. In particular the lattice P48nis the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.

Involutive Residuated Lattices Based on Modular and Distributive Lattices

An involutive residuated lattice (IRL) is a lattice-ordered monoid possessing residual operations and a dualizing element. We show that a large class of self-dual lattices may be endowed with an IRL structure, and give examples of lattices which fail to admit IRLs with natural algebraic conditions. A classification of all IRLs based on the modular lattices M n is provided.

Unimodular embeddings

In 1994 Morris Newman showed that a unimodular quadratic form on a lattice over a principal ideal domain can be represented by a triple-diagonal matrix of a rather special form, though the matrices associated with a given lattice in this way are generally not unique. The present paper considers positive definite unimodular lattices over the integers, and it begins the exploration of connections between those special matrix representations for a given lattice and the isometry class of that lattice.

Short-range entangled bosonic states with chiral edge modes andTduality of heterotic strings

We consider states of bosons in two dimensions that do not support anyons in the bulk, but nevertheless have stable chiral edge modes that are protected even without any symmetry. Such states must have edge modes with central charge $c=8k$ for integer $k$. While there is a single such state with $c=8$, there are, naively, two such states with $c=16$, corresponding to the two distinct even unimodular lattices in 16 dimensions. However, we show that these two phases are the same in the bulk, which is a consequence of the uniqueness of signature $(8k+n,n)$ even unimodular lattices. The bulk phases are stably equivalent, in a sense that we make precise. However, there are two different phases of the edge corresponding to these two lattices, thereby realizing a novel form of the bulk-edge correspondence. Two distinct fully chiral edge phases are associated with the same bulk phase, which is consistent with the uniqueness of the bulk since the transition between them, which is generically first order, can occur purely at the edge. Our construction is closely related to $T$ duality of toroidally compactified heterotic strings. We discuss generalizations of these results.

On extremal even unimodular 72-dimensional lattices

By computer search we show that the lattice Γ \Gamma of Nebe (2012) is the unique extremal even unimodular 72-dimensional lattices that can be constructed as proposed by Griess (2010).

A Classification of Unimodular Lattice Wiretap Codes in Small Dimensions

Lattice coding over a Gaussian wiretap channel, where an eavesdropper listens to transmissions between a transmitter and a legitimate receiver, is considered. A new lattice invariant called the secrecy gain is used as a code design criterion for wiretap lattice codes since it was shown to characterize the confusion that a chosen lattice can cause at the eavesdropper: the higher the secrecy gain of the lattice, the more confusion. In this paper, secrecy gains of extremal odd unimodular lattices as well as unimodular lattices in dimension <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , 16 ≤ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ≤ 23, are computed, covering the four extremal odd unimodular lattices and all the 111 nonextremal unimodular lattices (both odd and even), providing thus a classification of the best wiretap lattice codes coming from unimodular lattices in dimension <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , 8 <; <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ≤ 23. Finally, to permit lattice encoding via Construction A, the corresponding error correction codes of the best lattices are determined.

Some extremal self-dual codes and unimodular lattices in dimension 40

In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a classification of extremal odd unimodular lattices in dimension 40 with shadows having 80 vectors of norm 2 through their relationships with extremal doubly even self-dual codes of length 40.

Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of Thetanullwerte and the amazing Klein formula

Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g>=4, a new class of vector-valued modular forms, defined on the Teichmuller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus 4 is proportional to the square root of the products of Thetanullwerte \chi_{68}, which is a proof of the recently rediscovered Klein `amazing formula'. Furthermore, it turns out that the coefficients of such a quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian locus, implying new theta relations involving the latter, \chi_{68} and the theta series corresponding to the even unimodular lattices E_8\oplus E_8 and D_{16}^+. We also find, for g=4, a functional relation between the singular component of the theta divisor and the Riemann period matrix.

On a Conjecture by Belfiore and Solé on Some Lattices

The point of this paper is to show that the secrecy function attains its maximum at <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$y=1$</tex> </formula> on all known extremal unimodular lattices and on some possibly existing even extremal unimodular lattices. This is a special case of a conjecture by Belfiore and Solé. Furthermore, we will give a very simple method to verify or disprove the conjecture on any given unimodular lattice.

Even unimodular Lorentzian lattices and hyperbolic volume

Abstract. We compute the hyperbolic covolume of the automorphism group of each even unimodular Lorentzian lattice. The result is obtained as a consequence of a previous work with Belolipetsky, which uses Prasad's volume to compute the volumes of the smallest hyperbolic arithmetic orbifolds.

ON THE CLASSIFICATION OF EVEN UNIMODULAR LATTICES WITH A COMPLEX STRUCTURE

This paper classifies the even unimodular lattices that have a structure as a Hermitian [Formula: see text]-lattice of rank r ≤ 12 for rings of integers in imaginary quadratic number fields K of class number 1. The Hermitian theta series of such a lattice is a Hermitian modular form of weight r for the full modular group, therefore we call them theta lattices. For arbitrary imaginary quadratic fields we derive a mass formula for the principal genus of theta lattices which is applied to show completeness of the classifications.

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).

An even unimodular 72-dimensional lattice of minimum 8

An even unimodular 72-dimensional lattice Γ having minimum 8 is constructed as a tensor product of the Barnes lattice and the Leech lattice over the ring of integers in the imaginary quadratic number field with discriminant −7. The automorphism group of Γ contains the absolutely irreducible rational matrix group (PSL2(7)× SL2(25)) : 2.

Optimal self-dual [formula omitted]-codes and a unimodular lattice in dimension 41

For lengths up to 47 except 37, we determine the largest minimum Euclidean weight among all Type I Z4-codes of that length. We also give the first example of an optimal odd unimodular lattice in dimension 41 explicitly, which is constructed from some Type I Z4-code of length 41.

An optimal odd unimodular lattice in dimension 72

It is shown that if there is an extremal even unimodular lattice in dimension 72, then there is an optimal odd unimodular lattice in that dimension. Hence, the first example of an optimal odd unimodular lattice in dimension 72 is constructed from the extremal even unimodular lattice which has been recently found by G. Nebe.

Entropy and escape of mass for SL3(ℤ)\ SL3(ℝ)

We study the relation between measure theoretic entropy and escape of mass for the case of a singular diagonal flow on the moduli space of three-dimensional unimodular lattices.