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

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...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Ángel Luis Muñoz Castañeda, Noemí DeCastro-García, Miguel V. Carriegos
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	convolutional codes representations rings of modular integers Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/ea0d4cea1fa040aa87e33d1a22896aed
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:ea0d4cea1fa040aa87e33d1a22896aed
record_format	dspace
spelling	oai:doaj.org-article:ea0d4cea1fa040aa87e33d1a22896aed2021-11-25T18:17:36ZOn the State Approach Representations of Convolutional Codes over Rings of Modular Integers10.3390/math92229622227-7390https://doaj.org/article/ea0d4cea1fa040aa87e33d1a22896aed2021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2962https://doaj.org/toc/2227-7390In 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.Ángel Luis Muñoz CastañedaNoemí DeCastro-GarcíaMiguel V. CarriegosMDPI AGarticleconvolutional codesrepresentationsrings of modular integersMathematicsQA1-939ENMathematics, Vol 9, Iss 2962, p 2962 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	convolutional codes representations rings of modular integers Mathematics QA1-939
spellingShingle	convolutional codes representations rings of modular integers Mathematics QA1-939 Ángel Luis Muñoz Castañeda Noemí DeCastro-García Miguel V. Carriegos On the State Approach Representations of Convolutional Codes over Rings of Modular Integers
description	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.
format	article
author	Ángel Luis Muñoz Castañeda Noemí DeCastro-García Miguel V. Carriegos
author_facet	Ángel Luis Muñoz Castañeda Noemí DeCastro-García Miguel V. Carriegos
author_sort	Ángel Luis Muñoz Castañeda
title	On the State Approach Representations of Convolutional Codes over Rings of Modular Integers
title_short	On the State Approach Representations of Convolutional Codes over Rings of Modular Integers
title_full	On the State Approach Representations of Convolutional Codes over Rings of Modular Integers
title_fullStr	On the State Approach Representations of Convolutional Codes over Rings of Modular Integers
title_full_unstemmed	On the State Approach Representations of Convolutional Codes over Rings of Modular Integers
title_sort	on the state approach representations of convolutional codes over rings of modular integers
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/ea0d4cea1fa040aa87e33d1a22896aed
work_keys_str_mv	AT angelluismunozcastaneda onthestateapproachrepresentationsofconvolutionalcodesoverringsofmodularintegers AT noemidecastrogarcia onthestateapproachrepresentationsofconvolutionalcodesoverringsofmodularintegers AT miguelvcarriegos onthestateapproachrepresentationsofconvolutionalcodesoverringsofmodularintegers
_version_	1718411413192441856

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

Ejemplares similares