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 paper, we introduce the notion of complete j-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 T and show that complete T-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 j-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 F_13 and F_16 and we also give constructions for (2,1,3) complete 4-MDP convolutional codes over F_128 obtained by a randomized computer search.Comment: 2

Necessary field size and probability for MDP and complete MDP convolutional codes

It has been shown that maximum distance profile (MDP) convolutional codes have optimal recovery rate for windows of a certain length, when transmitting over an erasure channel. In addition, the subclass of complete MDP convolutional codes has the ability to reduce the waiting time during decoding. Since so far general constructions of these codes have only been provided over fields of very large size, there arises the question about the necessary field size such that these codes could exist. In this paper, we derive upper bounds on the necessary field size for the existence of MDP and complete MDP convolutional codes and show that these bounds improve the already existing ones. For some special choices of the code parameters, we are even able to give the exact minimum field size. Moreover, we derive lower bounds for the probability that a random code is MDP respective complete MDP

Weighted Reed-Solomon convolutional codes

In this paper we present a concrete algebraic construction of a novel class of convolutional codes. These codes are built upon generalized Vandermonde matrices and therefore can be seen as a natural extension of Reed-Solomon block codes to the context of convolutional codes. For this reason we call them weighted Reed-Solomon (WRS) convolutional codes. We show that under some constraints on the defining parameters these codes are Maximum Distance Profile (MDP), which means that they have the maximal possible growth in their column distance profile. We study the size of the field needed to obtain WRS convolutional codes which are MDP and compare it with the existing general constructions of MDP convolutional codes in the literature, showing that in many cases WRS convolutional codes require significantly smaller fields.Comment: 30 page