Applications of Gr¨ obner bases to the structural description and realization of multidimensional convolutional code
Chalie Charoenlarpnopparut · 2009
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 Grbases and m-D convolutional code is explored. Several applications of Gr ¨ obner 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 Gr ¨ obner 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.