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Construction and decoding of convolutional codes over the erasure channel

擦除通道上卷积码的构造和解码

基本信息

批准号：
392752124
负责人：
Professorin Dr. Julia Lieb
金额：
--
依托单位：
Institut für Mathematik
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2017
资助国家：
德国
起止时间：
2016-12-31 至 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/392752124?language=en
关键词：
Construction decoding convolutional codes over

项目摘要

Coding theory is dealing with error correction during data transmission using error correcting codes. The principle of an error correcting code is to add redundancy to a message, in order to make it possible to correct transmission errors or to reconstruct lost data.The aim of the project is to develop new constructions and efficient decoding algorithms for maximum distance profile (MDP) convolutional codes over the erasure channel. When using this channel, a transmitted symbol is either received correctly or gets lost. Moreover, special subclasses of MDP convolutional codes, the reverse-MDP and complete-MDP convolutional codes should be considered. These codes have additional properties, which are of advantage for decoding. The motivation for this project is provided by the necessity for decoding with least possible delay in applications like internet traffic and in particular, video streaming.In the first part of the project, I want to develop general constructions for reverse-MDP and complete-MDP convolutional codes, where for complete-MDP convolutional codes, one has to prove their existence for all code parameters first. Up to now there are only constructions for MDP convolutional codes over fields of large size, which makes decoding algorithms very complex. Therefore, another aim is to develop constructions of MDP convolutional codes for small field sizes and to obtain a bound for the necessary field size such that a MDP convolutional code exists.In the second part of the project, I plan to approach the development of efficient decoding algorithms for MDP convolutional codes over the erasure channel in two ways: firstly, by considering the parity-check matrix and secondly, by using the systems-theoretic representation of a convolutional code. For the second way, I will use that each convolutional code could be described by a linear system over a finite field.Finally, I will investigate if the class of MDP convolutional codes is capable to achieve Shannon capacity over the q-ary erasure channel without memory, i.e. the aim is to find codes with possibly large transmission rates and possibly small error probabilities. When using the above channel, symbols from a finite field with q elements are transmitted and the erasure probability of a symbols is independent of previous symbols.

编码理论是利用纠错码处理数据传输过程中的纠错问题。纠错码的原理是在信息中增加冗余，以便纠正传输错误或重建丢失的数据。该项目的目的是开发擦除信道上最大距离轮廓（MDP）卷积码的新结构和有效的解码算法。当使用此信道时，传输的符号要么被正确接收，要么丢失。此外，还应考虑MDP卷积码的特殊子类，即反向MDP卷积码和完全MDP卷积码。这些代码具有额外的属性，这是有利于解码。该项目的动机是在互联网流量，特别是视频流等应用中以尽可能少的延迟进行解码的必要性。在项目的第一部分，我想开发反向MDP和完整MDP卷积码的一般构造，对于完整MDP卷积码，必须首先证明其所有代码参数的存在。到目前为止，MDP卷积码的构造都是在大尺寸的域上进行的，这使得译码算法非常复杂。因此，另一个目标是开发小字段大小的MDP卷积码的结构，并获得必要的字段大小的界限，以便MDP卷积码存在。在该项目的第二部分，我计划以两种方式开发擦除信道上的MDP卷积码的有效解码算法：首先通过考虑奇偶校验矩阵，其次通过使用卷积码的系统理论表示。对于第二种方式，我将使用每个卷积码可以由有限域上的线性系统来描述。最后，我将研究这类MDP卷积码是否能够在没有记忆的q元擦除信道上实现香农容量，即目标是找到具有可能大的传输速率和可能小的错误概率的代码。当使用上述信道时，传输来自具有q个元素的有限域的符号，并且符号的擦除概率与先前的符号无关。