Complete j-MDP Convolutional Codes

Abstract

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 F13 and F16 and we also give constructions for (2, 1, 3) complete 4-MDP convolutional codes over F128 obtained by a randomized computer search.

A decoding algorithm for 2D convolutional codes over the erasure channel

“…Remark 1. A decoding algorithm for 1D convolutional codes over the erasure channel can also be found in [3]. This focuses more on decoding with low delay and how to restart the decoding if part of the sequence has to be declared as lost.…”

“…Here we concentrate on full recovery and formulate it in terms of the column distances of the code. Both Algorithm 1 of this paper and the algorithm in [3] have the advantage that they do the recovery of erasures in possibly small windows what reduces the decoding delay with respect to the decoding algorithm from [17] where windows of size (L + 1)n are considered, which causes a delay of L time instants.…”

Erasure decoding of convolutional codes using first-order representations

“…Recall that, when transmitting over this kind of channel, each symbol is either received correctly or is not received at all. The first decoding algorithm of convolutional codes over the erasure channel using parity-check matrices can be found in [20], variations of it in [1] or [11]. To investigate the capability of error correction of convolutional codes, it is necessary to define distance measures for these codes.…”

“…This implies that the decoding delay is always at least L. Moreover, it is computationally less complex and less costly to do several decoding steps in small windows than one decoding step in a larger window whose size is the sum of the sizes of the smaller windows since it is easier to solve several small than one large linear system of equations. The idea of improving this aspect of the algorithm from [20] is not new and can be found in several other papers (see [1,16]). In the following, we will illustrate the two main advantages that our algorithm has with respect to these papers.…”

“…Our algorithm exploits the ISO representation of a convolutional code via linear systems to recover the erasures in the received sequence. With respect to other erasure decoding algorithms for convolutional codes that can be found in the literature [1,16,20], our systems theoretic approach has the advantage that the computational effort and the decoding delay can be reduced.…”