Related papers: Line codes generated by finite Coxeter groups
In Index coding there is a single sender with multiple messages and multiple receivers each wanting a different set of messages and knowing a different set of messages a priori. The Index Coding problem is to identify the minimum number of…
The binary $k$-dimensional simplex code is known to be a $2^{k-1}$-batch code and is conjectured to be a $2^{k-1}$-functional batch code. Here, we offer a simple, constructive proof of a result that is "in between" these two properties. Our…
Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are…
We study the Picard groups of connected linear algebraic groups, and especially the subgroup of translation-invariant line bundles. We prove that this subgroup is finite over every global function field. We also utilize our study of these…
A rateless code-i.e., a rate-compatible family of codes-has the property that codewords of the higher rate codes are prefixes of those of the lower rate ones. A perfect family of such codes is one in which each of the codes in the family is…
Network coding theory studies the transmission of information in networks whose vertices may perform nontrivial encoding and decoding operations on data as it passes through the network. The main approach to deciding the feasibility of…
We construct complex root spaces remaining invariant under antilinear involutions related to all Coxeter groups. We provide two alternative constructions: One is based on deformations of factors of the Coxeter element and the other based on…
Lattices possess elegant mathematical properties which have been previously used in the literature to show that structured codes can be efficient in a variety of communication scenarios, including coding for the additive white Gaussian…
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 propose a generalized construction for binary polar codes based on mixing multiple kernels of different sizes in order to construct polar codes of block lengths that are not only powers of integers. This results in a multi kernel polar…
A new family of binary linear completely transitive (and, therefore, completely regular) codes is constructed. The covering radius of these codes is growing with the length of the code. In particular, for any integer r > 1, there exist two…
Random linear network codes can be designed and implemented in a distributed manner, with low computational complexity. However, these codes are classically implemented over finite fields whose size depends on some global network parameters…
A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some…
Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we…
The choice of an isomorphism, a duality, between a finite abelian group $A$ and its character group allows one to define dual codes of additive codes over $A$. Properties of dualities and dual codes are studied, continuing work of Delsarte…
This letter shows that linear Cellular Automata based on rules 90/150 generate all the solutions of linear difference equations with binary constant coefficients. Some of these solutions are pseudo-random noise sequences with application in…
Two classes of turbo codes over high-order finite fields are introduced. The codes are derived from a particular protograph sub-ensemble of the (dv=2,dc=3) low-density parity-check code ensemble. A first construction is derived as a…
A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates linear-programming based reception for coded modulation systems which use direct modulation mapping of coded…
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…
Error correction codes are a crucial part of the physical communication layer, ensuring the reliable transfer of data over noisy channels. The design of optimal linear block codes capable of being efficiently decoded is of major concern,…