有限环上线性码及其Gray象的应用研究

研究有限环上线性码的结构和性质特点，特别是利用Gray映射构造有限域上性能良好的纠错码一直是环上编码理论研究中的一个热点。首先，本项目将研究有限环上的等重码、2重量码的结构和性质，利用生成矩阵的结构和Pless等式探讨环上等重码和2重量码存在的充要条件。其次，利用Hensel引理、中国剩余定理以及傅里叶变换等思想方法建立有限环上任意长度的广义准扭码的结构理论，确定码的记数公式并给出算法。再次，研究非主理想环，如GF(p,m)+vGF(p,m)， GF(p,m)+vGF(p,m)+ v^2GF(p,m)上任意长度的线性码及其对偶码的MacWilliams恒等式、常循环自对偶码的生成多项式。最后，研究上述各种码的Gray映射象的结构特点，利用Gray映射设计搜索码的算法，以此尝试利用大型计算机搜索性能良好的纠错码。本课题的研究将对更深入的研究有限环上的纠错码理论及其在通信中的应用起到重要作用。

It has been a hot issue in the research of coding theory to explore the structures and properties of linear codes over finite rings, especially to construct good error-correcting codes over finite field by Gray map. Firstly, this programme studies the structures and properties of one-weight codes and 2-weight codes over finite rings,and uses the structures of generator matrix and Pless identities to probe into the necessary and sufficient condition for one-weight and two-weight codes over the ring to exist; Secondly, this programme uses Hense Lemma, Chinese remainder Theorem and discrete Fouirer transform methods to construct the structural theory of generalized quasi-quisted codes of arbitary length over the finite ring,and determine enumeration of codes as well as give an algorithm to count them. Thirdly,this programme studies MacWilliams identities of linear codes and its dual codes of arbitrary length as well as generator polynominals of constacyclic self-dual codes over non-principal ideal rings such as GF(p,m)+vGF(p,m)，GF(p,m)+vGF(p,m)+ v^2GF(p,m); Finally, this programme studies the structural properties of Grap images of the codes mentioned above, and uses Gray map to design algorithms to search codes,thus attempting to use maxicomputer to search good codes. This research will play a significant role in the further study of error-correcting codes over finite rings and its application in correspondence.