62 research outputs found
On minimality of convolutional ring encoders
Convolutional codes are considered with code sequences modeled as semi-infinite Laurent series. It is well known that a convolutional code C over a finite group G has a minimal trellis representation that can be derived from code sequences. It is also well known that, for the case that G is a finite field, any polynomial encoder of C can be algebraically manipulated to yield a minimal polynomial encoder whose controller canonical realization is a minimal trellis. In this paper we seek to extend this result to the finite ring case G = ℤ_{p^r} by introducing a so-called "p-encoder". We show how to manipulate a polynomial encoding scheme of a noncatastrophic convolutional code over ℤ_{p^r} to produce a particular type of p-encoder ("minimal p-encoder") whose controller canonical realization is a minimal trellis with nonlinear features. The minimum number of trellis states is then expressed as p^γ, where γ is the sum of the row degrees of the minimal p-encoder. In particular, we show that any convolutional code over ℤ_{p^r} admits a delay-free p-encoder which implies the novel result that delay-freeness is not a property of the code but of the encoder, just as in the field case. We conjecture that a similar result holds with respect to catastrophicity, i.e., any catastrophic convolutional code over ℤ_{p^r} admits a noncatastrophic p-encoder. © 2009 IEEE
On the State Approach Representations of Convolutional Codes over Rings of Modular Integers
[EN] 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.S
Realization of 2D convolutional codes of rate 1/n by separable Roesser models
In this paper, two-dimensional convolutional codes constituted by sequences in where is a finite field, are considered. In particular, we restrict to codes with rate and we investigate the problem of minimal dimension for realizations of such codes by separable Roesser models. The encoders which allow to obtain such minimal realizations, called R-minimal encoders, are characterized
Input-state-output representations and constructions of finite-support 2D convolutional codes
Two-dimensional convolutional codes are considered, with codewords having compact support indexed in N^2 and taking values in F^n, where F is a finite field. Input-state-output representations of these codes are introduced and several aspects of such representations are discussed. Constructive procedures of such codes with a designed distance are also presented. © 2010 AIMS-SDU
State-Space Realizations of Periodic Convolutional Codes
Convolutional codes are discrete linear systems over a finite field and can be defined as F[d]-modules, where F[d] is the ring of polynomials with coefficient in a finite field F. In this paper we study the algebraic properties of periodic convolutional codes of period 2 and their representation by means of input-state-output representations. We show that they can be described as F[d2]-modules and present explicit representation of the set of equivalent encoders. We investigate their state-space representation and present two different but equivalent types of state-space realizations for these codes. These novel representations can be implemented by realizing two linear time-invariant systems separately and switching the input (or the output) that is entering (or leaving) the system. We investigate their minimality and provide necessary and also sufficient conditions in terms of the reachability and observability properties of the two linear systems involved. The ideas presented here can be easily generalized for codes with period larger than 2.This work was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (FCT-Fundaçao para a Ciência e a Tecnologia) within project UIDB/04106/2020. It was also partially supported by Base Funding (UIDB/00147/2020) and Programmatic Funding (UIDP/00147/2020) of the Systems and Technologies Center - SYSTEC - funded by national funds through the FCT/MCTES (PIDDAC). The work of the second author was partially supported by Spanish grants PID2019-108668GB-I00 of the Ministerio de Ciencia e Innovación of the Gobierno de España and VIGROB-287 of the Universitat d'Alacant
The dual of convolutional codes over
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 codes from finite fields to finite rings have attracted much
attention in recent years due to fact that they are the most appropriate codes
for phase modulation. However convolutional codes over finite rings are more
involved and not fully understood. Many results and features that are
well-known for convolutional codes over finite fields have not been fully
investigated in the context of finite rings. In this paper we focus in one of
these unexplored areas, namely, we investigate the dual codes of convolutional
codes over finite rings. In particular we study the p-dimension of the dual
code of a convolutional code over a finite ring. This contribution can be
considered a generalization and an extension, to the rings case, of the work
done by Forney and McEliece on the dimension of the dual code of a
convolutional code over a finite field.Comment: submitte
Convolutional codes under control theory point of view. Analysis of output-observability
In this work we make a detailed look at the algebraic structure of convolutional codes using techniques
of linear systems theory. The connection between these concepts help to better understand the properties of convo-
lutional codes, in particular the concepts of controllability and observability of linear systems can be translated into
the context of convolutional codes relating these properties with the noncatastrophicity of the codes. We examine
the output-observability property and we give conditions for this property.Postprint (published version
Matrix fraction descriptions in convolutional coding
Doutoramento em MatemáticaOs objectos de estudo desta tese são os códigos convolucionais sobre um
corpo, constituídos por sequências com suporte compacto à esquerda.
Aplicando a abordagem comportamental à teoria dos sistemas, é obtida uma
nova definição de código convolucional baseada em propriedades estruturais
do próprio código.
Os codificadores e os formadores de síndrome de um código convolucional
são, respectivamente, as representações de imagem e as representações de
núcleo do código. As suas estruturas e propriedades são estudadas, utilizando
representações matriciais fraccionárias (RMF's). Seguidamente, são
analisados os codificadores e formadores de síndrome minimais de um código
convolucional, sendo apresentada uma parametrização simples das suas
RMF's. Mostra-se também como obter todos os codificadores minimais de um
código convolucional por aplicação de realimentação estática do estado e précompensação.
De modo análogo, obtêm-se todos os formadores de síndrome
minimais utilizando injecção da saída e pós-compensação.
Finalmente, estudam-se os codificadores desacoplados de um código
convolucional, que estão directamente ligados à sua decomposição.
Apresenta-se um algoritmo para determinação de um codificador desacoplado
maximal, que permitirá obter a decomposição máxima do código. Quando se
restringe a análise dos codificadores desacoplados aos minimais, obtém-se
um codificador canónico desacoplado e parametriza-se, utilizando RMF's,
todos os codificadores minimais que apresentam grau máximo de
desacoplamento.The objects of study of this thesis are the convolutional codes over a field,
constituted by left compact sequences.
To define a convolutional code we consider the behavioral approach to
systems theory, and present a new definition of convolutional code, taking into
account its structural properties.
Matrix Fractions Descriptions (MFD’s) are used as a tool for investigating the
structure of the encoders and the syndrome formers of a convolutional code,
which are, respectively, the image and the kernel representations of the code.
Next, we concentrate on the study of the minimal encoders and syndrome
formers, and obtain a simple parametrization of their MFD’s. We also show that
static feedback and precompensation allow to obtain all minimal encoders of
the code. The same is done for the minimal syndrome formers, using output
injection and postcompensation.
Finally, we analyse the decoupled encoders of a convolutional code, which are
associated with code decomposition. We provide an algorithm to determine a
maximally decoupled encoder, and, consequently, the finest decomposition of
the code. Restricting to minimal decoupled encoders, we first obtain a
canonical decoupled one, and parametrize, via MFD’s, all minimal decoupled
encoders realizing the finest decomposition of the code
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