On the State Approach Representations of Convolutional Codes over Rings of Modular Integers

Ángel Luis Muñoz Castañeda,Miguel V Carriegos,Noemí Decastro-García

doi:10.3390/math9222962

Ángel Luis Muñoz Castañeda, Miguel V Carriegos + Show 1 more

Open Access

https://doi.org/10.3390/math9222962

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Journal: Mathematics	Publication Date: Nov 20, 2021
Citations: 1	License type: CC BY 4.0

Affiliation: University of Leon

Abstract

In this study, we prove the existence of minimal first-order representations for convolutional codes with the predictable degree property over principal ideal artinian rings. Further, we prove that any such first-order representation leads to an input/state/output representation of the code provided the base ring is local. When the base ring is a finite field, we recover the classical construction, studied in depth by J. Rosenthal and E. V. York. This allows us to construct observable convolutional codes over such rings in the same way as is carried out in classical convolutional coding theory. Furthermore, we prove the minimality of the obtained representations. This completes the study of the existence of input/state/output representations of convolutional codes over rings of modular integers.

Highlights

The relation between convolutional codes and linear dynamical systems has been and still is largely studied
If we describe a convolutional code as a linear subspace, C ⊂ Fq (z)n, the linear system associated with the code is known as driving input/output representation ([1,2])
If we describe a convolutional code as a submodule, C ⊂ Fq [z]n, the coding dynamics can be modeled by a linear dynamical system known as input/state/output (I/S/O) representation [3,4,5,6], since k components of the output drive the remaining n − k components

Summary

Introduction

The relation between convolutional codes and linear dynamical systems has been and still is largely studied. One can define a convolutional code as a time-invariant complete behavior In such a case, there is a representation theory [7,8,9]. The mathematical formalism of the theory of convolutional codes over general rings is very similar to that of fields, their properties may be quite different, and they need to be studied for particular rings ([22,23,24]) Despite their importance, the extension of the relation between minimal I/S/O representations and convolutional codes to an arbitrary commutative ring may not be that close.

Convolutional Codes and Linear Systems

Algebraic Preliminaries about Convolutional Codes

A Review of the Representations of Convolutional Codes over Finite Fields

On the Predictable Degree Property of Polynomial Encoders

On the Existence of Representations over Principal Ideal Artinian Rings

Conclusions