Unimodular Lattices In Dimension Research Articles

Self-dual codes over [formula omitted] and s-extremal unimodular lattices

New s-extremal extremal unimodular lattices in dimensions 38, 40, 42 and 44 are constructed from self-dual codes over F5 by Construction A. In the process of constructing these codes, we obtain a self-dual [44,22,14] code over F5. In addition, the code implies a [43,22,13] code over F5. These codes have larger minimum weights than the previously known [44,22] codes and [43,22] codes, respectively.

On the distribution of angles between increasingly many short lattice vectors

Following Södergren, we consider a collection of random variables on the space Xn of unimodular lattices in dimension n: normalizations of the angles between the N=N(n) shortest vectors in a random unimodular lattice, and the volumes of spheres with radii equal to the lengths of these vectors. We investigate the expected values of certain functions (whose support depends on a parameter K=K(n)) evaluated at these random variables in the regime where K and N are allowed to tend to infinity with n at the rate KN=o(n1/6). Our main result is that as n⟶∞, these random variables exhibit a joint Poissonian and Gaussian behavior.

Construction of s-extremal optimal unimodular lattices in dimension 52

An s-extremal optimal unimodular lattice in dimension 52 is constructed for the first time. This lattice is constructed from a certain self-dual -code by Construction A. In addition, as neighbours of the lattice, two more s-extremal optimal unimodular lattices are constructed.

2- and 3-Modular lattice wiretap codes in small dimensions

A recent line of work on lattice codes for Gaussian wiretap channels introduced a new lattice invariant called secrecy gain as a code design criterion which captures the confusion that lattice coding produces at an eavesdropper. Following up the study of unimodular lattice wiretap codes [1], this paper investigates 2- and 3-modular lattices and compares them with unimodular lattices. Most even 2- and 3-modular lattices are found to have better performance, that is, a higher secrecy gain than the best unimodular lattices in dimension n, n is between 2 and 23. Odd 2-modular lattices are considered, too, and three lattices are found to outperform the best unimodular lattices.

Automorphisms of extremal unimodular lattices in dimension 72

The paper narrows down the possible automorphisms of extremal even unimodular lattices of dimension 72. With extensive computations in Magma[7] using the very sophisticated algorithm for computing class groups of algebraic number fields written by Steve Donnelly it is shown that the extremal even unimodular lattice Γ72 from [17] is the unique extremal even unimodular lattice of dimension 72 that admits a large automorphism, where a d×d matrix is called large, if its minimal polynomial has an irreducible factor of degree >d/2.

Extremal Unimodular Lattices in Dimension 36

New extremal odd unimodular lattices in dimension 36 are constructed. Some new odd unimodular lattices in dimension 36 with long shadows are also constructed.

A fourth extremal even unimodular lattice of dimension 48

We show that there is a unique extremal even unimodular lattice of dimension 48 which has an automorphism of order 5 of type 5-(8, 16)-8. Since the three known extremal lattices do not admit such an automorphism, this provides a new example of an extremal even unimodular lattice in dimension 48.

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.

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.

CONSTRUCTION OF SOME UNIMODULAR LATTICES WITH LONG SHADOWS

In this paper, we construct odd unimodular lattices in dimensions n = 36,37 having minimum norm 3 and 4s = n - 16, where s is the minimum norm of the shadow. We also construct odd unimodular lattices in dimensions n = 41,43,44 having minimum norm 4 and 4s = n - 24.

Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers

All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in $\mathbb{Q}(\sqrt{-3})$ are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3.

EISENSTEIN LATTICES, GALOIS RINGS AND QUATERNARY CODES

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.

On some self-dual codes and unimodular lattices in dimension 48

In this paper, binary extremal self-dual codes of length 48 and extremal unimodular lattices in dimension 48 are studied through their shadows and neighbors. We relate an extremal singly even self-dual [48, 24, 10] code whose shadow has minimum weight 4 to an extremal doubly even self-dual [48, 24, 12] code. It is also shown that an extremal odd unimodular lattice in dimension 48 whose shadow has minimum norm 2 relates to an extremal even unimodular lattice. Extremal singly even self-dual [48, 24, 10] codes with shadows of minimum weight 8 and extremal odd unimodular lattice in dimension 48 with shadows of minimum norm 4 are investigated.

Construction of new extremal unimodular lattices

In this paper we construct new extremal and optimal unimodular lattices in dimensions 36, 38, 42, 45, 52, 54, 60 and 68. We construct them in two ways: first in the case of dimensions congruent to 4 modulo 8 by construction B 3 followed by density doubling, generalizing the constructions of Sphere Packing Lattices and Groups (1988) 148 and Théorie des Nombres (1989) 772; and second by applying the well known Construction A to self-dual codes over GF(5) and to codes over the ring Z/25 Z. In particular the lattice in dimension 60, P 60 q , generalizes the construction of the lattice P 48 q . We also give the complete weight enumerator of the extended ternary quadratic residue code of length 60 and we provide a table of the best known unimodular lattices of dimensions up to 80.

Extremal odd unimodular lattices in dimensions 44, 46 and 47

In this note, extremal odd unimodular lattices in dimensions 44, 46 and 47 are constructed from self-dual codes over Z 4 and Z 6 by Construction A. The lattices in dimensions 46 and 47 seem to be the first explicit examples for such lattices.

A mass formula for unimodular lattices with no roots

We derive a mass formula fornn-dimensional unimodular lattices having any prescribed root system. We use Katsurada’s formula for the Fourier coefficients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32-dimensional lattices and odd unimodular lattices of dimensionn≤30n\leq 30. In particular, we find the mass of even unimodular 32-dimensional lattices with no roots, and the mass of odd unimodular lattices with no roots in dimensionn≤30n\leq 30, verifying Bacher and Venkov’s enumerations in dimensions 27 and 28. We also compute better lower bounds on the number of inequivalent unimodular lattices in dimensions 26 to 30 than those afforded by the Minkowski-Siegel mass constants.

Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over \Bbb Z 6

We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772–784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we correct an error on the condition for the minimum norm of the lattices of dimension a multiple of 12. As the results of our present research, extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time. It is shown that the unimodular lattices obtained by the method can be constructed from some self-dual ℤ6-codes. Then extremal self-dual ℤ6-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.

On the existence of extremal Type II codes over [formula omitted

Recently, there has been tremendous interest in self-dual codes over finite rings, specifically the rings Z 2k . In this note, the first examples of extremal Type II codes over Z 6 of length 32 are constructed. These give examples of extremal even unimodular lattices in dimension 32 by Construction A. A relationship between extremal Type II codes over Z 6 and binary extremal Type II codes is also investigated.