Related papers: On the Existence of Cyclic Lattice Codes
In the present paper we introduce and study a canonical ${\cal E}$-lattice structure on the set of element orders of some finite groups. We show that a finite abelian group is uniquely determined by this canonical ${\cal E}$-lattice.
In this paper, we study dilation of cyclic polytopes with the vertices defined by a generator of the simplest cubic fields. In particular, for a specific range of values, we give a precise number of the contained lattice points.
We prove the existence of a finitely dependent proper colouring of the integer lattice Z^d that is fully isometry-invariant in law, for all dimensions d. Previously this was known only for d=1, while only translation-invariant examples were…
A circulant-based spatially-coupled (SC) code is constructed by partitioning the circulants in the parity-check matrix of a block code into several components and piecing copies of these components in a diagonal structure. By connecting…
Cyclic codes are an important class of linear codes. Bounding the minimum distance of cyclic codes is a long-standing research topic in coding theory, and several well-known and basic results have been developed on this topic. Recently,…
We investigate group coding for arbitrary finite groups acting linearly on a vector space. These yield robust codes based on real or complex matrix groups. We give necessary and sufficient conditions for correct subgroup decoding using…
A method for finding an optimum $n$-dimensional commutative group code of a given order $M$ is presented. The approach explores the structure of lattices related to these codes and provides a significant reduction in the number of…
The problem of identifying whether the family of cyclic codes is asymptotically good or not is a long-standing open problem in the field of coding theory. It is known in the literature that some families of cyclic codes such as BCH codes…
Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In…
We present a systematic method for constructing lattice QCD operators for systems of an arbitrary number of particles with arbitrary momentum, spin, and internal quantum numbers. Explicit constructions are provided for one-, two-, three-,…
We construct multilevel lattice codes from multiquadratic number fields for the compound block-fading wiretap channel. More precisely, we specialize Construction $\pi_A$ over the ring of integers $\mathcal{O}_K$ and exploit rational primes…
For a pair of given binary perfect codes C and D of lengths t and m respectively, the Mollard construction outputs a perfect code M(C,D) of length tm + t + m, having subcodes C1 and D2, that are obtained from codewords of C and D…
Let $G$ be a finite group, $L_1(G)$ be its poset of cyclic subgroups and consider the quantity $\alpha(G)=\frac{|L_1(G)|}{|G|}$. The aim of this paper is to study the class $\cal{C}$ of finite nilpotent groups having…
In this paper, we propose to study and optimize a very general class of LDPC codes whose variable nodes belong to finite sets with different orders. We named this class of codes Hybrid LDPC codes. Although efficient optimization techniques…
This work shows a study about the structure of the cycles contained in a Minimal Strong Digraph (MSD). The structure of a given cycle is determined by the strongly connected components (or strong components, SCs) that appear after…
We consider $d$-dimensional lattice polytopes $\Delta$ with $h^*$-polynomial $h^*_\Delta=1+h_k^*t^k$ for $1<k<(d+1)/2$ and relate them to some abelian subgroups of $\SL_{d+1}(\C)$ of order $1+h_k^*=p^r$ where $p$ is a prime number. These…
Ordinary Coincidence Site Lattices (CSLs) are defined as the intersection of a lattice $\Gamma$ with a rotated copy $R\Gamma$ of itself. They are useful for classifying grain boundaries and have been studied extensively since the mid…
Using the classification and description of the structure of bisimple monogenic orthodox semigroups obtained in \cite{key10}, we prove that every bisimple orthodox semigroup generated by a pair of mutually inverse elements of infinite order…
The aim of the present paper is to study isomorphisms of canonical ${\cal E}$-lattices. Some interesting results are obtained in the particular case of isomorphisms between two subgroup ${\cal E}$-lattices.
Low density parity check (LDPC) lattices are obtained from Construction D' and a family of nested binary LDPC codes. We consider an special case of these lattices with one binary LDPC code as underlying code. This special case of LDPC…