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Given a lattice $\Lambda$ in a locally compact abelian group $G$ and a measurable subset $\Omega$ with finite and positive measure, then the set of characters associated to the dual lattice form a frame for $L^2(\Omega)$ if and only if the…
Linear complementary dual (LCD) codes are linear codes which intersect their dual codes trivially, which have been of interest and extensively studied due to their practical applications in computational complexity and information…
The {\it linear representation} of a subset of a finite projective space is an incidence system of affine points and lines determined by the subset. In this paper we use character theory to show that the rank of the incidence matrix has a…
We present the Probabilistic Context Neighborhood model designed for two-dimensional lattices as a variation of a Markov Random Field assuming discrete values. In this model, the neighborhood structure has a fixed geometry but a variable…
We describe and analyze sparse graphical code constructions for the problems of source coding with decoder side information (the Wyner-Ziv problem), and channel coding with encoder side information (the Gelfand-Pinsker problem). Our…
A set of linearly constrained permutation matrices are proposed for constructing a class of permutation codes. Making use of linear constraints imposed on the permutation matrices, we can formulate a minimum Euclidian distance decoding…
We prove a gluing theorem for linearised vacuum gravitational fields in Bondi gauge on a class of characteristic hypersurfaces in static vacuum $(n+1)$-dimensional backgrounds with cosmological constant $ \Lambda \in \mathbb{R}$, $n\ge 4$.…
A novel construction of lattices is proposed. This construction can be thought of as a special class of Construction A from codes over finite rings that can be represented as the Cartesian product of $L$ linear codes over…
Linear codes are widely studied in coding theory as they have nice applications in distributed storage, combinatorics, lattices, cryptography and so on. Constructing linear codes with desirable properties is an interesting research topic.…
We investigate random spatially coupled low-density parity-check (SC-LDPC) code ensembles over finite fields. Under different variable-node edge-spreading rules, the random Tanner graphs of several coupled ensembles are defined by multiple…
This paper establishes information-theoretic limits in estimating a finite field low-rank matrix given random linear measurements of it. These linear measurements are obtained by taking inner products of the low-rank matrix with random…
Most multi-dimensional (more than two dimensions) lattice partitions only form additive quotient groups and lack multiplication operations. This prevents us from constructing lattice codes based on multi-dimensional lattice partitions…
By using totally isotropic subspaces in an orthogonal space Omega^{+}(2i,2), several infinite families of packings of 2^k-dimensional subspaces of real 2^i-dimensional space are constructed, some of which are shown to be optimal packings. A…
This paper supplies additions to our paper in Linear Algebra Appl. 510 (2016) 395--420 on integral spans of tight frames in Euclidean spaces. In that previous paper, we considered the case of an equiangular tight frame (ETF), proving that…
The paper presents a method to generate some families of linear codes over finite fields of characteristics greater than two in the widest class due to the size of Grassmann manifold, i.e. when the dimension is equal to codimension. Our…
We introduce a class of countable groups by some abstract group-theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing $\ell^2$-Betti numbers that…
Triangulations are important objects of study in combinatorics, finite element simulations and quantum gravity, where its entropy is crucial for many physical properties. Due to their inherent complex topological structure even the number…
A lattice of integers is the collection of all linear combinations of a set of vectors for which all entries of the vectors are integers and all coefficients in the linear combinations are also integers. Lattice reduction refers to the…
We establish an upper bound on the number of real multiquadratic fields that admit a universal quadratic lattice of a given rank, or contain a given amount of indecomposable elements modulo totally positive units, obtaining density zero…
In this paper, we study the shortest $t$-dimensional hull embeddings of linear codes in both Euclidean and Hermitian cases, extending the existing research on the shortest LCD and self-orthogonal embeddings to arbitrary hull dimensions and…