| With the fast development of information technology, it is necessary for us to look for a stable and fast way of transmitting information. As for different model of channel, people put forward various kinds of coding scheme to approximate Shannon limits. Among those codes, there are many well know codes such as Reed-Solomon code, Reed-Muller code and so on, as well as some seminal and under fast development codes such as network code, space-time code and so on. They hold their advantages over others when the environment is specified. Thus, they fail to substitute for each other. This dissertation mainly focus on the theoretical problems inside these different kind of codes. Our contribution includes following several points.1. Network code in multi-source network. Chapter2,3discuss some properties of multi-source network. As we know, the linear network codes can reach the capacity of the network in single-source networks. Howev-er, this property fails to be generalized to multi-source networks. Besides this, we know little about the multi-source networks. Those we have known almost contradict the conclusion made in single-source network-s. In other words, We investigate the multi-source networks based on conterexample. For those we have already known, we can claim that the Shannon inequality is not sufficient to determine the capacity of networks, nonlinear network codes can reach a larger capacity than linear network codes and so on. We construct a class of networks with specified proper-ties. We show that their capacity can not be achieved if the size of their finite field is less than the given number n. Of course, these networks are byproduct of our construction. We generalize the property proposed by [30] to network with any number of sources. They all keep the property. In Chapter3, we further give the construction of matroid network. They hold better property than the original one. For example, they can reflect properties of matroid and match the corresponding matroid very well.2. The security problem of convolutional network code. In Chapter4, we generalize the security problem of network codes. The security problem in acyclic networks is extended to cyclic networks. They are required to keep the strongly secure property or weakly secure property as theirs in acyclic network. To meet this requirement, we introduce the invariant factor theorem in commutative algebra. Thanks to this theorem, we successfully prove the existence of security in cyclic network. In particular, if the size of finite field is sufficiently large, we can do little change to original coding scheme. In stead, we only do linear combination on source nodes. As for those small size of finite field, we need to construct a new secure convolutional code.3. The construction of locally decodable code. In Chapter5, we investigate the construction of recently popular locally decodable code. Although it is not a high rate code, it can be applied to data storage and cryptograph. One of its good feature is that there is no need for the knowledge of the whole codeword when we want to decode a bit message. Reed-Muller code is an important class of locally decodable codes. The new coming construction of locally decodable code relies on the matching vectors. It generates a subexponential length locally decodable codes. When the query times is constant, it beats the Reed-Muller codes. We show a best known optimal matching vectors.4-The construction of complex orthogonal space-time code. In chap-ter6, we investigate the orthogonal space-time code which has important application in wireless communication system. It can achieve large code gain and diversity gain. To reach maximal code rate and minimal delay, it is necessary to design a theoretical optimal complex orthogonal space-time code. Many scientists have proved its theoretical bound. They also put forward some constructions based on algorithm or recursiveness. However, these construction is not explicit and lose the beautiful math-ematical structure. In chapter6, we give a class of complex orthogonal space-time codes which can achieve maximal rate and minimal delay. Our construction is explicit and can indicate the exact representation of entry in each position. They are also the first known explicit representation of complex orthogonal space-time code. |
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