Algebraic decoder of multidimensional convolutional code: constructive algorithms for determining syndrome decoder and decoder matrix based on Gröbner basis

Authors:
P. Jangisarakul;C. Charoenlarpnopparut
Affiliations:
School of Information, Computer and Communication Technology, Sirindhorn International Institute of Technology, Thammasat University, Klong Luang, Pathum-Thani, Thailand 12121;School of Information, Computer and Communication Technology, Sirindhorn International Institute of Technology, Thammasat University, Klong Luang, Pathum-Thani, Thailand 12121
Venue:
Multidimensional Systems and Signal Processing
Year:
2011

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Abstract

A representation of an multidimensional (m-D) convolutional encoder is analogous to the representation of a transfer function for a MIMO m-D FIR system. The encoder matrix is usually not square and thus finding its inverse (decoder matrix) typically employs the Moore-Penrose generalized inverse. However, the result may not be FIR (polynomial matrix) even if the generator matrix is a polynomial matrix. In this paper a constructive algorithm for computing the FIR pseudo inverse, based on the usage of Gröbner basis is presented along with detailed examples. The result obtained can be parameterized to cover the class of all possible FIR inverses. In addition, by using the computation method of syzygy with the Gröbner basis module, the syndrome matrix for a given m-D convolutional encoder is shown. Furthermore, the theory of Gröbner basis is applied to solve the algebraic syndrome decoder problems using the maximum likelihood (nearest neighborhood) criteria and the procedure for 2-D convolutional code error correction is proposed. Despite the complication of the decoding process, the proposed method is the only error correcting decoder for multidimensional convolutional code available to date.