Narrow Results

In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field ${𝒦}=\mathbb{Q}({\varsigma }_8)$ over $\mathbb{Q}$. We employ binary codes to generate a lattice over $\mathbb{Q}({\varsigma }_8)$, which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels.

We derive the equivalent energy of a square lattice that either deforms into the three- dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy–Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomistic models.

Let be a Kac–Moody Lie algebra. We give an interpretation of Tits' associated group functor using representation theory of and we construct a locally compact "Kac–Moody group" G over a finite field k. Using (twin) BN-pairs (G,B,N) and (G,B^-,N) for G we show that if k is "sufficiently large", then the subgroup B^- is a non-uniform lattice in G. We have also constructed an uncountably infinite family of both uniform and non-uniform lattices in rank 2. We conjecture that these form uncountably many distinct conjugacy classes in G. The basic tool for the construction of non-uniform lattices in rank 2 is a spherical Tits system for G which we also construct.

This paper presents three main ideas. They are the Metatheorem, the lattice embedding for sets, and the lattice embedding for algebras.

The Metatheorem allows you to convert existing theorems about classical subsets into corresponding theorem about fuzzy subsets. The concept of a fuzzyfiable operation on a powerset is defined. The main result states that any implication or identity which can be stated using fuzzyfiable operations is true about fuzzy subsets if and only if it is true about classical subsets.

The lattice embedding theorem for sets shows that for any set X, there is a set Y such that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of Y. In fact it is further proved that if X is infinite, then we can choose Y = X and get the surprising result that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of X itself. The idea is illustrated with an example explicitly showing how the lattice of fuzzy subsets of the closed unit interval 𝕀 = [0,1] embeds into the lattice of classical subsets of 𝕀.

The lattice embedding theorem for algebras shows that under certain circumstances the lattice of fuzzy subalgebras of an algebra A embeds into the lattice of classical subalgebras of a closely related algebra A′. The following sample use of this embeding theorem is given. It is a well known fact that the lattice of normal subgroups of a group is a modular lattice. The embeding theorem is used here to conclude that lattice of fuzzy normal subgroups of a group is a modular lattice too.

We take a Galois ring $GR(p^m,g)$ and discuss about the self-dual codes and its properties over the ring. We will also describe the relationship between Clifford-Weil group and Jacobi forms by constructing the invariant polynomial ring with the complete weight enumerator.

In this paper, we showed that any clone of operations preserving a nontrivial $n$-equivalence on a finite set is categorically equivalent to the clone of operations preserving a nontrivial $n$-equivalence on a set having the cardinality of $3$.

In this paper, we consider an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-m})$ with $-m\equiv 3$ (mod $4$). In particular, we study the ring of integers corresponding to the field $K$ and visualize the form of ${𝜖}_K/p{𝜖}_K$. We also consider lattices over the ring of integers ${𝜖}_K$ and discuss the theta series to see its relation with the weight enumerator. As a consequence, we will see how the theta series differs for different $m$ and $m^{\prime }$.

Reducing the size of a logic circuit through lattice identities is an important and well-studied discrete optimization problem. In this paper, we consider a related problem of integrating several circuits into a single hypercircuit using the recently developed concept of lattice hyperterms. We give a combinatorial algorithm for integrating k-out-of-n symmetrical diagrams which play an important role in reliability theory. Our results show that the integration can reduce the number of circuit gates by more than twice.

We propose in this work a Sieve algorithm that we called OrthogonalInteger sieve algorithm for some orthogonal integer lattices and particularly the case of integer lattices $\mathrm{\Lambda }\subset {\mathbb{Z}}^n$, root lattices of type $A_n$ ($n\ge 1$) and of type $D_n$ ($n\ge 2$). In these cases, we use the famous $LLL$ algorithm to find the shortest vector of these lattices. Indeed, in general, a sieve algorithm builds a list of short random vectors which are not necessarily in the lattice, and try to produce short lattice vectors by taking linear combinations of the vectors in the list. But in our case, we built a list of short vectors in the lattice. From the first column of the $LLL$-reduced basis of the considered basis, we have the list of at least $n$ and at most $2^n$ short vectors for the general case (where $n$ is the dimension of the lattice) of orthogonal integer lattices $\mathrm{\Lambda }\subset {\mathbb{Z}}^n$. For the lattices ${\mathbb{Z}}^n$, $A_n$ ($n\ge 1$) and $D_n$ ($n\ge 2$), we have, respectively, $2n$, $n(n+1)$ and $2n(n-1)$ short vectors. The proposed sieve algorithm for integer lattice ${\mathbb{Z}}^n$ runs in space $O(2n)$ and the OrthogonalInteger sieve algorithm performs $O(n{2}^n)$ arithmetic operations and is polynomial in space.