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 article, we introduce the notion of complete 3-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 3-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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">13</sub> and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">16</sub> and we also give constructions for (2, 1, 3) complete 4-MDP convolutional codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">128</sub> obtained by a randomized computer search.
Highlights
C ONVOLUTIONAL codes are especially suitable for transmitting over an erasure channel, which is the most used channel in multimedia traffic
They introduced the subclass of reverse maximum distance profile (MDP) convolutional codes, which could recover more erasure patterns since they are suitable for sliding backwards along the sequence
Complete MDP convolutional codes, which are again a subclass of reverse MDP convolutional codes, have the additional benefit that they can correct even more erasure patterns than reverse MDP convolution codes, e.g. there is less waiting time when a large burst of erasures occurs and no correction is possible for some time
Summary
C ONVOLUTIONAL codes are especially suitable for transmitting over an erasure channel, which is the most used channel in multimedia traffic. [10] contains a description of all (2, 1, 1) MDP, reverse MDP and complete MDP convolutional codes over the smallest possible field F3. We continue this work giving a description of all (2, 1, 2) complete MDP convolutional codes with minimal field size. We give a complete description of all (2, 1, 2) complete MDP convolutional codes over the smallest possible fields, namely F13 and F16 These results were obtained with the help of a computer search using the mathematical software Maple. In Appendix, we include an illustrative sample of the computer programs we used
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