Minimal Trellis Construction for Finite Support Convolutional Ring Codes

Kuijper, Margreta; Pinto, Raquel

doi:10.1007/978-3-540-87448-5_11

Cited by 9 publications

(8 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 2 Conform [1,15,21,27,30] we have decided to define our codes as finite support convolutional codes. There exists however a considerable body of literature in which code sequences are semi-infinite Laurent series [7,11,18,23,24].…”

Section: Block Codesmentioning

confidence: 99%

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ Z p r

Napp

Pinto

Toste

2016

Des. Codes Cryptogr.

Self Cite

View full text Add to dashboard Cite

Resumo Maximum Distance Separable (MDS) convolutional codes are characterized through the property that the free distance meets the generalized Singleton bound. The existence of free MDS convolutional codes over Z p r was recently discovered in [26] via the Hensel lift of a cyclic code. In this paper we further investigate this important class of convolutional codes over Z p r from a new perspective. We introduce the notions of p-standard form and roptimal parameters to derive a novel upper bound of Singleton type on the free distance. Moreover, we present a constructive method for building general (non necessarily free) MDS convolutional codes over Z p r for any given set of parameters.

show abstract

Section: Block Codesmentioning

confidence: 99%

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ Z p r

Napp

Pinto

Toste

2016

Des. Codes Cryptogr.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Evidently lm(f ) ≤ max 1≤i≤N ;ai =0 (lm(a i ) lm(v i )), so that (12) implies that equality holds. This proves the p-PLM property.…”

Section: Special Case IImentioning

confidence: 99%

“…Indeed, the p-PLM property clearly implies the p-predictable degree property of [15]. Thus one of the applications where a minimal TOP Gröbner p-basis can be used is in convolutional coding over Z p r : a minimal TOP Gröbner p-basis then serves as a minimal p-encoder of a convolutional code over Z p r in the terminology of [12,13]. Applications for which the p-PLM property is particularly useful are parametrizations for minimal interpolation-type problems, as illustrated in the next subsection.…”

Section: An Application Over Z P Rmentioning

confidence: 99%

Minimal Gröbner bases and the predictable leading monomial property

Kuijper

Schindelar

2011

Linear Algebra and its Applications

View full text Add to dashboard Cite

We focus on Gröbner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the "predictable leading monomial (PLM) property" that is shared by minimal Gröbner bases of modules in F[x] q , no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property (= row reducedness), a terminology introduced by Forney in the 70's. Because of the presence of zero divisors, minimal Gröbner bases over a finite ring of the type Zpr (where p is a prime integer and r is an integer > 1) do not necessarily have the PLM property. In this paper we show how to derive, from an ordered minimal Gröbner basis, a so-called "minimal Gröbner p-basis" that does have a PLM property. We demonstrate that minimal Gröbner p-bases lend themselves particularly well to derive minimal realization parametrizations over Zpr . Applications are in coding and sequences over Zpr . * M. Kuijper is with the

show abstract

“…Recent papers [13], [16], [14], [15] consider behaviors over the ring Z p r , where p is a prime integer and r is a positive integer. In these papers the theory of [17] is put to work to extend the above two minimal realization problems to systems over the ring Z p r .…”

Section: Introductionmentioning

confidence: 99%

Gröbner bases and behaviors over finite rings

Kuijper

Schindelar

2009

Proceedings of the 48h IEEE Conference on Decision and Control (CDC) Held Jointly With 2009 28th Chinese Control Conference

Self Cite

View full text Add to dashboard Cite

For several decades Gröbner bases have proved useful tools for different areas in system theory, particularly multidimensional system theory. These areas range from controller design to minimal realizations of linear systems over fields. In this paper we focus on the univariate case and identify the so-called "predictable leading monomial property" as a property of a minimal Gröbner basis that is crucial in many of these areas. The property is stronger than "row reducedness". We revisit the recently developed theory of [17] in which row reducedness is extended to polynomial matrices over the finite ring Zpr (with p a prime integer and r a positive integer), which find applications in error control coding over Zpr . We recast the ideas of [17] in the more general setting of Gröbner bases and derive new results on how to use minimal Gröbner bases to achieve the predictable leading monomial property over Zpr . A major advantage of the Gröbner approach is that computational packages are available to compute a minimal Gröbner basis over Zpr , such as the SINGULAR computer algebra system. Another advantage of the Gröbner approach is its generality with respect to the choice of ordering of polynomial vectors.

show abstract

Minimal Trellis Construction for Finite Support Convolutional Ring Codes

Cited by 9 publications

References 16 publications

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ Z p r

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ Z p r

Minimal Gröbner bases and the predictable leading monomial property

Gröbner bases and behaviors over finite rings

Contact Info

Product

Resources

About

Minimal Trellis Construction for Finite Support Convolutional Ring Codes

Cited by 9 publications

References 16 publications

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ Z p r

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$ Z p r

Minimal Gröbner bases and the predictable leading monomial property

Gr&#x00F6;bner bases and behaviors over finite rings

Contact Info

Product

Resources

About

Gröbner bases and behaviors over finite rings