Applications of Gröbner bases to the structural description and realization of multidimensional convolutional code

Chalie Charoenlarpnopparut

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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School of Information, Computer and Communication Technology, Sirindhorn International Institute of Technology, Thammasat University, Klong Luang, Pathum Thani 12121, Thailand

* Corresponding author, E-mail: chalie@siit.tu.ac.th

Received 25 Feb 2008, Accepted 4 Mar 2009