Multidimensional convolutional code: progresses and bottlenecks

Charoenlarpnopparut, Chalie; Tantaratana, S.

doi:10.1109/iscas.2003.1205112

Cited by 5 publications

(3 citation statements)

References 13 publications

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“…Now, since g 1 and g j are polynomials and r j does not divide g j , then r 1 /r j must be a polynomial, which means that g 1 |g j , which contradicts the assumption that g 1 and g j have no common divisors except for units. Therefore vIm{G}■ A more general procedure for constructing parity check matrices for codes of rate k/n is given in [5], which uses Groebner bases.…”

Section: Definitionmentioning

confidence: 99%

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Ball Codes - Two-Dimensional Tail-Biting Convolutional Codes

Alfandary

Raphaeli

2010

2010 IEEE Global Telecommunications Conference GLOBECOM 2010

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In this paper we investigate a new class of codes, the 2-D tail-biting convolutional codes (2-D TBCCs). The class of two-dimensional convolutional codes (2-D CCs) is a littleresearched subject in coding theory, and tail-biting versions were hardly mentioned, unless they can be represented as a product of two 1-D codes. These codes have interesting geometry since they are the 2-D analog of the 1-D TBCC which their graph is a ring. The result being a code invariant to shifts in 2-D space. We apply algebraic methods in order to find bijective encoders, create parity check matrices and inverse encoders. Next, we discuss minimum distance and weight distribution properties of these codes. We observe that some of these codes exhibit very good codes performance. We then present several novel iterative suboptimal algorithms for soft decoding 2-D CCs, which are based on belief propagation and generalized belief propagation. The results show that the suboptimal algorithms achieve respectable results, in some cases coming as close as 0.4dB from optimal (maximum-likelihood) decoding. I. INTRODUCTIONMulti-dimensional convolutional codes (m-D CCs, where m stands for the dimension) extend the notion of 1-D CCs [1] to m-D information sequences and generator polynomials. However, in contrast to the one dimensional case, little research has been done in the field of m-D CCs. Significantly, the algebraic theory for m-D CCs has been laid out by Foransini and Valcher [2] and Weiner [3]. Lobo et al. [4]have also investigated the subject, concentrating on a subfamily of these codes dubbed "Locally Invertible m-D CCs". Some progress in the field was also made by Charoenlarpnopparut et al.,[5], who suggest a method to realize a 2-D CC encoder, and a construction for its parity check matrix.We believe 2-D TBCCs to be interesting from an academic point of view, as these codes were largely unexplored to-date, and are a non-trivial generalization of 1-D TBCCs. From the results point of view, we have shown that among 2-D TBCCs there are good codes for medium sized blocks compared to existing codes, so these codes have practical potential. We hope these codes will find applications as error correcting codes for medium length blocks, for example in wireless applications.Tail biting [1] is a well known technique to efficiently convert CCs into block codes. In this paper we extend the notion to m-D CCs, and derive some algebraic properties for them with focus on rate 1/n codes. We show: how to test whether an encoder is bijective, how to test whether it has a polynomial inverse; and how to construct a polynomial inverse and a parity check matrix in case they exist. Then we use a tree-search algorithm to find the distance properties of

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Section: Definitionmentioning

confidence: 99%

“…Lobo et al [4] have also investigated the subject, concentrating on a sub-family of these codes dubbed "Locally Invertible m-D CCs". Charoenlarpnopparut et al, [5] also contributed to the field by suggesting a method for realizing a 2D CC encoder, and constructing its parity check matrix .…”

Section: Introductionmentioning

confidence: 99%

Ball Codes - Two-Dimensional Tail-Biting Convolutional Codes

Alfandary

Raphaeli

2010

2010 IEEE Global Telecommunications Conference GLOBECOM 2010

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“…In the later part of the paper, the problem of minimal realization [14][15][16] is considered. Partial solutions by direct sequential search and algebraic reduction using the Gröbner basis for modules are provided.…”

Section: Introductionmentioning

confidence: 99%

Untitled

Charoenlarpnopparut¹

2009

ScienceAsia

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ABSTRACT:Over the past few years, multidimensional convolutional code has become an emerging area of research in the signal processing community. While one-dimensional convolutional code and its variants have been thoroughly understood, the m-D counterpart still lacks unified notation and efficient encoding/decoding implementation. Here, the strong link between the theory of Gröbner bases and m-D convolutional code is explored. Several applications of Gröbner bases to the characterization of m-D convolutional encoders are proposed. Furthermore, the more practical problem of minimal encoder realization is discussed and an algebraic algorithm based on the use of Gröbner bases is provided. From the implementation point of view, the syndrome decoder is currently the only means for decoding m-D convolutional code. A constructive method for computing the syndrome decoding matrix using the theory of syzygy modules is proposed.

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Algebraic approach to reduce the number of delay elements in the realization of multidimensional convolutional code

Charoenlarpnopparut

Tantaratana

The 2004 47th Midwest Symposium on Circuits and Systems, 2004. MWSCAS '04.

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Multidimensional convolutional code: progresses and bottlenecks

Cited by 5 publications

References 13 publications

Ball Codes - Two-Dimensional Tail-Biting Convolutional Codes

Ball Codes - Two-Dimensional Tail-Biting Convolutional Codes

Untitled

Algebraic approach to reduce the number of delay elements in the realization of multidimensional convolutional code

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