LDPC lattices were the first family of lattices that equipped with iterative decoding algorithms under which they perform very well in high dimensions. In this paper, we introduce quasi cyclic low density parity check (QC-LDPC) lattices as a special case of LDPC lattices with one binary QC-LDPC code as their underlying code. These lattices are obtained from Construction A of lattices providing ...

A semimodular lattice L of finite length will be called an almost-geometric lattice, if the order J(L) of its nonzero join-irreducible elements is a cardinal sum of at most two-element chains. We prove that each finite distributive lattice is isomorphic to the lattice of congruences of a finite almost-geometric lattice.

We present a duality theorem for bounded lattices that improves and strengthens Urquhart's topological representation for lattices. Rather than using maximal, disjoint lter-ideal pairs, as Urquhart does, we use all disjoint lter-ideal pairs. This allows not only for establishing a bijective correspondance between lattices and a certain kind of doubly ordered Stone Spaces (Urquhart), but for a f...

Principal lattices in the plane are distributions of points particularly simple to use Lagrange, Newton or Aitken-Neville interpolation formulas. Principal lattices were generalized by Lee and Phillips, introducing three-pencil lattices, that is, points which are the intersection of three lines, each one belonging to a different pencil. In this contribution, a semicubical parabola is used to co...

Let p be an odd prime. It is known that the symplectic group Sp2n(p) has two (algebraically conjugate) irreducible representations of degree (pn +1)/2 realized over Q(√ p), where = (−1)(p−1)/2. We study the integral lattices related to these representations for the case pn ≡ 1 mod 4. (The case pn ≡ 3 mod 4 has been considered in a previous paper.) We show that the class of invariant lattices co...

The classification problem is solved for some type of nonlinear lattices. These lattices are closely related to the lattices of Ruijsenaars–Toda type and define the Bäcklund auto-transformations for the class of two-component hyperbolic systems.

This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to isomorphism) in PU(9, 1) of smallest Euler–Poincaré characteristic amongst all no...

Discretizations of the Laplacian operator on non-hypercubical lattices are discussed in a systematic approach. It is shown that order a2 errors always exist for discretizations involving only nearest neighbors. Among all lattices with the same density of lattice sites, the hypercubical lattices always have errors smaller than other lattices with the same number of spacetime dimensions. On the o...

In this short note we show two results concerning sampling translation invariant subspaces of L(R) on unions of lattices. The first result shows that the sampling transform on a union of lattices is a constant times an isometry if and only if the sampling transform on each individual lattice is so. The second result demonstrates that the sampling transforms of two unions of lattices on two band...

In this paper, we study the separtion axioms T0, T1, T2 and T5/2 on topological and semitopological residuated lattices and we show that they are equivalent on topological residuated lattices. Then we prove that for every infinite cardinal number α, there exists at least one nontrivial Hausdorff topological residuated lattice of cardinality α. In the follows, we obtain some conditions on (semi)...