Maximum distance separable 2D convolutional codes

• Main Author: Climent, J.-J.
• Publication Date: 2016
• Other Authors: Napp, D., Perea, C., Pinto, Raquel
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10773/15106
• topic: 2D convolutional code; Circulant Cauchy matrix; Generalized Singleton bound; Maximum distance separable code; Superregular matrix

Maximum distance separable (MDS) block codes and MDS 1D convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1D convolutional codes of a certain rate and degree, respectively. In this paper, we generalize this concept to the class of 2D convolutional codes. For that, we introduce a natural bound on the distance of a 2D convolutional code of rate $k/n$ and degree $delta $ , which generalizes the Singleton bound for block codes and the generalized Singleton bound for 1D convolutional codes. Then, we prove the existence of 2D convolutional codes of rate $k/n$ and degree $delta $ that reach such bound when $n geq k (({(lfloor ({delta }/{k}) rfloor + 2)(lfloor ({delta }/{k}) rfloor + 3)})/{2})$ if $k {nmid } delta $ , or $n geq k (({(({delta }/{k}) + 1)(({delta }/{k}) + 2)})/{2})$ if $k mid delta $ , by presenting a concrete constructive procedure.