Complete j-MDP Convolutional Codes

Maximum distance profile (MDP) convolutional codes have been proven to be very suitable for transmission over an erasure channel. In addition, the subclass of complete MDP convolutional codes has the ability to restart decoding after a burst of erasures. However, there is a lack of constructions of these codes over fields of small size. In this article, we introduce the notion of complete <inline-formula> <tex-math notation="LaTeX">$j$ </tex-math></inline-formula>-MDP convolutional codes, which are a generalization of complete MDP convolutional codes, and describe their decoding properties. In particular, we present a decoding algorithm for decoding erasures within a given time delay <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula> and show that complete <inline-formula> <tex-math notation="LaTeX">$T$ </tex-math></inline-formula>-MDP convolutional codes are optimal for this algorithm. Moreover, using a computer search with the MAPLE software, we determine the minimal binary and non-binary field size for the existence of (2,1,2) complete <inline-formula> <tex-math notation="LaTeX">$j$ </tex-math></inline-formula>-MDP convolutional codes and provide corresponding constructions. We give a description of all (2,1,2) complete MDP convolutional codes over the smallest possible fields, namely <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{13}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{16}$ </tex-math></inline-formula> and we also give constructions for (2,1,3) complete 4-MDP convolutional codes over <inline-formula> <tex-math notation="LaTeX">$\mathbb F_{128}$ </tex-math></inline-formula> obtained by a randomized computer search.