Feedback equivalence of convolutional codes over finite rings

DeCastro-García, Noemí

doi:10.1515/math-2017-0124

Cited by 8 publications

(5 citation statements)

References 33 publications

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“…We require the encoding map to be injective and therefore focus on free submodules of Z p r [d] n . We note that different definitions have been considered in the literature, see for instance [4,10,11,21]. The nonfree case lies beyond the scope of this work but it can also be treated using the theory of p-basis and p-generating sequences, see for instance [11,12,18,21].…”

Section: Distance Properties Of Free Convolutional Codes Over Z P Rmentioning

confidence: 99%

Noncatastrophic convolutional codes over a finite ring

2021

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Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized as polynomial left prime matrices. In this paper, we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in the finite ring [Formula: see text]. In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring [Formula: see text]. The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated [Formula: see text]-modules. We introduce the notion of [Formula: see text]-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on [Formula: see text] is.

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Section: Distance Properties Of Free Convolutional Codes Over Z P Rmentioning

confidence: 99%

Noncatastrophic convolutional codes over a finite ring

2021

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“…In particular, the algebraic study of linear systems in the state-space approach [3] deals with linear systems defned on algebras and modules over a commutative ring [4,5]. Tis approach has been used recently in the feld of convolutional codes [6][7][8][9][10]. Convolutional codes are in fact error-correcting codes over a fnite feld F defned as vector subspaces of F(z) n , where F (z) is the feld of rationals which are realized as linear control systems over F .…”

Section: Introductionmentioning

confidence: 99%

Natural Factorization of Linear Control Systems through Parallel Gathering of Simple Systems

Carriegos

2023

Journal of Mathematics

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Linear systems over vector spaces and feedback morphisms form an additive category taking into account the parallel gathering of linear systems. This additive category has a minimal exact structure and thus a notion of simple systems as those systems have no subsystems apart from zero and themselves. The so-called single-input systems are proven to be exactly the simple systems in the category of reachable systems over vector spaces. The category is also proven to be semisimple in objects because every reachable linear system is decomposed in a finite parallel gathering of simple systems. Hence, decomposition result is fulfilled for linear systems and feedback morphisms, but category of reachable linear systems is not abelian semisimple because it is not balanced and hence fails to be abelian. Finally, it is conjectured that the category of linear systems and feedback actions is in fact semiabelian; some threads to find the result and consequences are also given.

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“…Linear control systems over commutative rings are a generalization of linear control systems used in the study of parametric families of linear systems, 1–3 or in the study of convolutional codes over finite rings 4,5 …”

Section: Introductionmentioning

confidence: 99%