Related papers: Linear representations of convolutional codes over…
In this paper, we develop a representation-theoretic formulation of discrete-time linear systems. We show that such systems are naturally viewed as representations of time groups acting on vector spaces, thereby endowing the state space…
An important class of codes widely used in applications is the class of convolutional codes. Most of the literature of convolutional codes is devoted to con- volutional codes over finite fields. The extension of the concept of convolutional…
We consider the category of linear relations over an arbitrary commutative ring, and identify it as a subcategory of the category of Kronecker representations. We observe that this subcategory forms a definable, faithful and hereditary…
In this paper, in the first we give definitions of some classes of division rings which strictly contain the class of centrally finite division rings. One of our main purpose is to construct non-trivial examples of rings of new defined…
In this paper, we develop the theory of convolutional codes over finite commutative chain rings. In particular, we focus on maximum distance profile (MDP) convolutional codes and we provide a characterization of these codes, generalizing…
In this paper, we employ the linear systems representation of a convolutional code to develop a decoding algorithm for convolutional codes over the erasure channel. We study the decoding problem using the state space description and this…
We study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. We construct a canonical correspondence between the irreducible representations of…
A general method for constructing convolutional codes from units in Laurent series over matrix rings is presented. Using group ring as matrix rings, this forms a basis for in-depth exploration of convolutional codes from group ring…
We construct, for any finite commutative ring $R$, a family of representations of the general linear group $\mathrm{GL}_n(R)$ whose intertwining properties mirror those of the principal series for $\mathrm{GL}_n$ over a finite field.
Faithful representations of regular $\ast$-rings and modular complemented lattices with involution within orthosymmetric sesquilinear spaces are studied within the framework of Universal Algebra. In particular, the correspondence between…
Observable convolutional codes defined over Zpr with the Predictable Degree Property admit minimal input/state/output representations that preserve structural properties under scalar restriction. We make use of this fact to present…
After giving an overview of the existing theory regarding the periods of sequences defined by linear recurrences over finite fields, we give explicit descriptions of the sets of periods that arise if one considers all sequences over…
We describe the "generic" part of the character ring of general linear groups over a finite field in terms of quiver representations.
Linear codes are widely employed in communication systems, consumer electronics, and storage devices. All linear codes over finite fields can be generated by a generator matrix. Due to this, the generator matrix approach is called a…
We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units.…
In this paper, we describe linear and cyclic codes over the rings of the form $R_{s,p}=\mathbb{Z}_{p}[u]/\left( f\left(u\right) /\left( u-s\right) \right)$, where $p$ is a prime number and $f\left( u\right) =u^{p}-u$, with $s\in…
This paper defines a linear representation for nonlinear maps $F:\mathbb{F}^n\rightarrow\mathbb{F}^n$ where $\mathbb{F}$ is a finite field, in terms of matrices over $\mathbb{F}$. This linear representation of the map $F$ associates a…
We propose a binary representation of categorical values using a linear map. This linear representation preserves the neighborhood structure of categorical values. In the context of evolutionary algorithms, it means that every categorical…
In this paper, we prove the existence of a first-order definition of the polynomial ring over a nonprincipal ultraproduct of finite fields of unbounded cardinalities in its fraction field by a universal-existential formula in the language…
We consider an arbitrary representation of the additive group over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants.