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

Column distances of convolutional codes over Z_p^r

Maximum distance profile codes over finite nonbinary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of p-encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over Zpr is presented.publishe

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

Constructing strongly-MDS convolutional codes with maximum distance profile

This paper revisits strongly-MDS convolutional codes with maximum distance profile (MDP). These are (non-binary) convolutional codes that have an optimum sequence of column distances and attains the generalized Singleton bound at the earliest possible time frame. These properties make these convolutional codes applicable over the erasure channel, since they are able to correct a large number of erasures per time interval. The existence of these codes have been shown only for some specific cases. This paper shows by construction the existence of convolutional codes that are both strongly-MDS and MDP for all choices of parameters