Series concatenation of 2D convolutional codes by means of input-state-output representations

ABSTRACT

In this paper, we investigate the properties of two-dimensional (2D) convolutional codes which are obtained from series concatenation of two 2D convolutional codes. For this purpose, we confine ourselves to dealing with finite-support 2D convolutional codes and make use of the Fornasini–Marchesini input-state-output (ISO) model representations. Within these ISO representations, we study when the structural properties of modal reachability and modal observability of the two given ISO representations carry over to the resulting 2D convolutional code. Moreover, we provide necessary conditions for obtaining a systematic concatenated convolutional code. Finally, we present a lower bound on its free distance.

KEYWORDS:

• 2D convolution codes
• series concatenation
• reachability
• observability

Acknowledgments

The authors would like to thank the reviewers for their valuable comments and suggestions which led to significant improvements.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the second, third and fourth authors was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA); The Portuguese Foundation for Science and Technology (FCT – Fundação para a Ciência e a Tecnologia) [grant number UID/MAT/04106/2013].