On duals and parity-checks of convolutional codes over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math>

Oued, Mohammed El; Napp, Diego; Pinto, Raquel; Toste, Marisa

doi:10.1016/j.ffa.2018.08.012

Cited by 6 publications

(4 citation statements)

References 13 publications

(14 reference statements)

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“…where the coefficients a i are in Z p r and only finitely coefficients with negative indices may be nonzero. It can be shown [4] that there always exists a polynomial matrix H(D) such that C = {w(D) ∈ Z n p r ((D)) : H(D)w(D) = 0} and therefore C ⊂ ker H(D) ⊂ Z n p r [D] as in described in (1). Moreover, H(D) is the smallest observable convolutional code containing C. For the sake of simplicity we consider only observable convolutional codes and develop are algorithms in terms of a parity-check polynomial matrix representing the code.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…The fact that these steps yield the set of all solution follows from[11, Theorem 3]. Let C ⊂ Z 8 [D] be the convolutional code with parity-check matrix H(D) =H 0 + H 1 D + H 2 D 2 ∈ Z 8 [D] where H 0 = is easy to check that w(D) = w 0 + w 1 D + w 2 D 2 + w 3 D 3 with w 0 = [5, 5, 0, 6, 0], w 1 = [6,6,4,3,6], w 2 = [2, 1, 1, 2, 0] and w 3 = [2, 6, 4, 0, 0] is a codeword of C. Assume that one receives w…”

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confidence: 99%

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List decoding of Convolutional Codes over integer residue rings

Lieb¹,

Napp²,

Pinto³

2020

Preprint

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where Z p r [D] stands for the ring of polynomials with coefficients in Z p r . In this paper we study the list decoding problem of these codes when the transmission is performed over an erasure channel, that is, we study how much information one can recover from a codeword w ∈ C when some of its coefficients have been erased. We do that using the p-adic expansion of w and particular representations of the parity-check matrix of the code. From these matrix representations we recursively select certain equations that w must satisfy and have only coefficients in the field p r−1 Z p r . This yields a step by step procedure to obtain a list of possible codewords for a given corrupted codeword w. We show that such an algorithm actually computes all possible erased coordinates, that is, it provides a minimal list with the closest codewords to the vector w. Mathematically, this problem amounts to determine the set of all possible solutions of a set of linear equations over Z p r [D] that can be represented by a matrix with Toeplitz structure.

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Section: Preliminary Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

List decoding of Convolutional Codes over integer residue rings

Lieb¹,

Napp²,

Pinto³

2020

Preprint

Self Cite

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“…Therefore, they are very suitable for streaming applications [8,18,19,25]. In the sequel, we follow the module-theoretic approach to convolutional codes as it was described in [8,12,31,15,33]…”

Section: Convolutional Codesmentioning

confidence: 99%

Systematic maximum sum rank codes

Almeida

Martínez-Peñas

Napp

2020

Finite Fields and Their Applications

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In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.

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“…Although 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.…”

Section: Introductionmentioning

confidence: 99%

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

2021

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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.

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On duals and parity-checks of convolutional codes over Zpr

Cited by 6 publications

References 13 publications

List decoding of Convolutional Codes over integer residue rings

List decoding of Convolutional Codes over integer residue rings

Systematic maximum sum rank codes

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

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