Superregular Lower Triangular Toeplitz Matrices for Low Delay Wireless Streaming

Hansen, Jens Christian; Østergaard, Jan; Kudahl, Johnny; Madsen, John H.

doi:10.1109/tcomm.2017.2712651

Cited by 13 publications

(12 citation statements)

References 25 publications

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“…We prove that this problem can be reduced to the construction of certain classes of matrices called superregular matrices (see Theorem 8) and present a concrete class of such matrices. The problem of deriving superregular matrices to build convolutional codes has become an active area of research, see for instance [3,6,12], and the results presented in this section extend the known results on this topic.…”

Section: Introductionsupporting

confidence: 64%

“…Since no column distance can achieve a value greater than the Singleton-type upper bound in (12), there must exist an integer L for which the bound ( 16) could be attained for all j ≤ L and it is a strict upper bound for j > L. It is a matter of straightforward computations to verify that this value is (see [13] for more details):…”

Section: Mrd Convolutional Codes: a Matrix Characterizationmentioning

confidence: 99%

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A new rank metric for convolutional codes

Almeida

Napp

2020

Des. Codes Cryptogr.

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Let F[D] be the polynomial ring with entries in a finite field F. Convolutional codes are submodules of F[D] n that can be described by left prime polynomial matrices. In the last decade there has been a great interest in convolutional codes equipped with a rank metric, called sum rank metric, due to their wide range of applications in reliable linear network coding. However, this metric suits only for delay free networks. In this work we continue this thread of research and introduce a new metric that overcomes this restriction and therefore is suitable to handle more general networks. We study this metric and provide characterizations of the distance properties in terms of the polynomial matrix representations of the convolutional code. Convolutional codes that are optimal with respect to this new metric are investigated and concrete constructions are presented. These codes are the analogs of Maximum Distance Profile convolutional codes in the context of network coding. Moreover, we show that they can be built upon a class of superregular matrices, with entries in an extension field, that preserve their superregularity properties even after multiplication with some matrices with entries in the ground field.

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Section: Introductionsupporting

confidence: 64%

Section: Mrd Convolutional Codes: a Matrix Characterizationmentioning

confidence: 99%

A new rank metric for convolutional codes

Almeida

Napp

2020

Des. Codes Cryptogr.

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“…In any case, the main problem of all these general constructions is that they require impractically large finite fields. For this reason, most of the optimal constructions of convolutional codes presented over finite fields of reasonable size are found via computer search and limited to small parameters, see for instance [3,8,9,12]. In this section we present concrete examples of superregular matrices, of given parameters and finite fields, that satisfy conditions of Theorem 6 and therefore yield m-MSR convolutional codes.…”

Section: Reducing the Field Size Of M-msr 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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“…The decoder tries to recover w i up to a given instant i + T and if this is not possible it outputs a list with the closest vectors at time instant T + i. The parameter T is called the delay constraint and represents the maximum delay the receiver can tolerate to retrieve w i , see [1,9,7]. For the sake of simplicity it will be assumed T ≤ ν, where ν is the degree of H(D).…”

Section: A Decoding Algorithm For Erasuresmentioning

confidence: 99%

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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Superregular Lower Triangular Toeplitz Matrices for Low Delay Wireless Streaming

Cited by 13 publications

References 25 publications

A new rank metric for convolutional codes

A new rank metric for convolutional codes

Systematic maximum sum rank codes

List decoding of Convolutional Codes over integer residue rings

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