A fundamental study of generalized compound-error-correcting-codes and their decoding algorithms

在本研究中，我们引入了一个新的概念，即发生在通信信道中的误差。这种误差在本研究中被称为复合误差，它由随机误差和突发误差两种误差组成。在此基础上，讨论了接收空间复合错误的译码原理。此外，我们还讨论了一些复合纠错码及其译码算法。特别地，我们将复合误差的概念扩展到擦除，也就是说，我们将两种类型的擦除，随机擦除和突发擦除，视为擦除。然后在接收空间中引入一种新的复合距离，该复合距离不满足数学距离的条件之一真三角不等式。但可以证明，在进行有界距离译码时，利用新的复合距离可以从理论上纠正复合擦除和复合错误。从这个关键定理出发，我们可以积极地考虑复合擦除码和复合纠错码的具体译码方法。我们证明了迭代码可以用作复合擦除和复合纠错码，其中迭代码是已知乘积码的广义版本。在此基础上，提出了迭代码的译码算法。所提出的译码算法可以在迭代码最小复合距离的一半的理论范围内纠正复合擦除和复合误差。此外，在未来的工作中，我们研究了几种网络码作为复合擦除和复合误差概念的应用。具体地，我们提出了鲁棒网络码和安全网络码的编码方法及其算法。我们估计了它们的网络代码的理论性质。

In this research we introduce a new concept of error which occurs in the communication channel. The error is called a compound error in this research, and the error consists of two types of errors which are random-errors and burst-errors. After that, we discuss a principle of decoding compound errors in the received space. Moreover, we discuss some compound-error-correcting-codes and their decoding algorithms. In particular, we extend the concept of compound error to erasures, that is, we consider two types of erasures, random-erasures and burst-erasures, as erasures. Then we introduce a new compound distance, which can not satisfy the true triangle inequation which is one of conditions for a mathematical distance, in the received space. But it can be proved that we can theoretically correct compound-erasures and compound-errors when we execute a bounded-distance decoding by using the new compound distance. From this key theorem, we can positively consider concrete decoding methods for compound-erasure and compound-error correcting codes. We show that iterated codes are able to be used as compound-erasure and compound-error correcting codes, where the iterated code is a generalized version of the well-known product code. After that, we propose the decoding algorithm for the iterated codes. The proposed decoding algorithm can correct compound-erasures and compound-errors up to theoretical bound which is the half of minimum compound distance of the iterated code. Moreover, for future works, we researched several network codes as applications of concepts of compound-erasures and compound-errors. Concretely, we proposed coding methods and their algorithms about robust network codes and secure network codes. And we estimated theoretical properties of their network codes.