Algebraic lower bounds on the free distance convolutional codes

Lally, Kristine

doi:10.1109/isit.2005.1523471

Cited by 3 publications

(5 citation statements)

References 19 publications

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“…This may cause an infinite number of fails in the decoding, which is undesired. Viewing convolutional codes as linear codes over F q (x) leads to codewords with infinite weight, which can not occur in practice and there is no reason to use this as the definition (see [8,21]). Moreover, again due to practical purposes, finite weight codewords which are causal are of interest ( [21,29]).…”

Section: Convolutional Codesmentioning

confidence: 99%

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Multidimensional Quasi-Cyclic and Convolutional Codes

Güneri̇

Özkaya

2016

IEEE Trans. Inform. Theory

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We introduce multidimensional generalizations of quasi-cyclic codes and investigate their algebraic properties as well as their links to multidimensional convolutional codes. We call these generalized codes n-dimensional quasi-cyclic (QnDC) codes. We provide a concatenated structure for QnDC codes in the sense that they can be decomposed into shorter codes over extensions of their base field. This structure allows us to prove that these codes are asymptotically good. Then, we extend the relation between quasi-cyclic and convolutional codes to multidimensional case. Lally has shown that the free distance of a convolutional code is lower bounded by the minimum distance of an associated quasi-cyclic code. We show that a QnDC code can be associated to a given nD convolutional code. Moreover, we prove that the relation between distances of convolutional and quasicyclic codes extend to a class of 1-generator 2D convolutional codes and the associated Q2DC codes. Along the way, an alternative new description of noncatastrophic polynomial encoders is given for 1-generator 1D convolutional codes and a sufficient condition for noncatastrophic nD polynomial encoders is obtained for 1-generator nD convolutional codes.

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Section: Convolutional Codesmentioning

confidence: 99%

“…Quasi-cyclic codes are naturally related to convolutional codes. It has been shown by Lally that the free distance of a convolutional code can be lower bounded by the minimum distance of an associated QC code (see [21]).…”

Section: Introductionmentioning

confidence: 99%

Multidimensional Quasi-Cyclic and Convolutional Codes

Güneri̇

Özkaya

2016

IEEE Trans. Inform. Theory

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“…i∈[m B ) be a codeword of the product code A ⊗ B, where each row is a codeword of A and each column is a codeword of B. The entry m i,j is the coefficient c µ(i,j) of the codeword i c i X i as in (8). In order to prove that A ⊗ B is -quasi-cyclic it is sufficient to show that a shift by positions of a codeword serialized to a univariate polynomial by (9)…”

Section: Quasi-cyclic Product Codementioning

confidence: 99%

“…we obtain an -quasi-cyclic shift of the univariate codeword obtained by ( 8) and (9). Instead of representing a codeword of A ⊗ B as one univariate polynomial as in (8), we want to represent it as univariate polynomials as defined in (1).…”

Section: Quasi-cyclic Product Codementioning

confidence: 99%

Construction of Quasi-Cyclic Product Codes

Zeh¹,

Ling²

2015

Preprint

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Linear quasi-cyclic product codes over finite fields are investigated. Given the generating set in the form of a reduced Gröbner basis of a quasi-cyclic component code and the generator polynomial of a second cyclic component code, an explicit expression of the basis of the generating set of the quasicyclic product code is given. Furthermore, the reduced Gröbner basis of a one-level quasi-cyclic product code is derived.

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“…II. BACKGROUND In this section, we give some background concerning classical convolutional codes, following [7,Chapter 14] and [10].…”

Section: Introductionmentioning

confidence: 99%

Quantum Convolutional Codes Derived from Generalized Reed-Solomon Codes

Aly

Klappenecker

Sarvepalli

2007

2007 IEEE International Symposium on Information Theory

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Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. A quantum Singleton bound for pure convolutional stabilizer codes is given. A familiy of quantum convolutional codes is derived from generalized Reed-Solomon codes. These codes are shown to be optimal with respect to the (quantum) Singleton bound.

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Algebraic lower bounds on the free distance convolutional codes

Abstract: Abstract-A new module structure for convolutional codes is introduced and used to establish further links with quasi-cyclic and cyclic codes.

Cited by 3 publications

References 19 publications

Multidimensional Quasi-Cyclic and Convolutional Codes

Multidimensional Quasi-Cyclic and Convolutional Codes

Construction of Quasi-Cyclic Product Codes

Quantum Convolutional Codes Derived from Generalized Reed-Solomon Codes

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