有限环上自对偶码的构造研究

Coding theory over finite rings is a hot research topic in the error-correcting theory. Self-dual codes over quaternary rings Z4 and F2+uF2 form an important class of linear codes. They are closely connected with other mathematical problems such as lattices, invariant theory, combinational designs and so on. The project mainly studies the construction of some long lengths self-dual codes with good parameters over the quaternary rings Z4 and F2+uF2. Firstly, by the connections of Z4 and F2+uF2 and their extension rings, we will research some algebraic structure of quadratic residue codes and their extension codes over the extension rings of Z4 and F2+uF2, and then construct some self-dual codes over the extension rings of Z4 and F2+uF2. Secondly, we will generalize some other well-known construction methods of self-dual codes over Z4 and F2+uF2 to the extension rings of Z4 and F2+uF2, and then construct some self-dual codes over the extension rings of Z4 and F2+uF2. Finally, we will give some useful Gray maps preserving the minimum distance and the self-duality from the extension rings of Z4 and F2+uF2 to Z4 and F2+uF2, which will lead to the construction of long lengths self-dual codes with good parameters over Z4 and F2+uF2. The project will make contributions to the development of the theory of error-correcting codes over finite rings and the application in mathematics fields.

有限环上的纠错码理论是一个热点研究问题。四元环Z4与F2+uF2上的自对偶码是重要的线性码类，与很多数学领域紧密相关如格理论、不变量理论、组合设计等。本项目将研究四元环Z4与F2+uF2上码长较大且性能良好的自对偶码的构造问题。首先，研究四元环Z4与F2+uF2的扩环上二次剩余码及其扩充码的代数结构，从而构造出Z4与F2+uF2的扩环上的自对偶码；其次，将Z4与F2+uF2上构造自对偶码的组合方法推广到Z4与F2+uF2的扩环上并由此构造Z4与F2+uF2的扩环上的自对偶码；最后，给出Z4与F2+uF2的扩环到Z4与F2+uF2上保距、保持自对偶性且使映射后的Gray象具有较大极小距离的Gray映射，利用Z4与F2+uF2的扩环上的自对偶码和Gray映射间接的构造Z4与F2+uF2上码长较大且性能良好的自对偶码。本项目将为有限环上纠错码理论的发展及其在数学领域中的应用做出一定的贡献。

本项目主要研究了有限环上几类线性码的代数结构及其在构造自四元对偶码、四元线性码、量子纠错码以及二元非线性码中的应用。研究了四元环Z4的某一个扩环上的自对偶循环码、拟循环码和广义拟循环码的代数结构，并由扩环到Z4的Gray映射构造了一些参数较好的四元自对偶码和线性码；研究了某几类有限非链环上二次剩余码、交错常循环码、常循环码的代数结构，并由有限非链环到有限域的Gray映射构造了一些参数较好的自对偶码和自正交码，进而利用量子纠错码的构造方法(CSS构造方法)构造了一些新的量子纠错码；研究了环Z4x(F2+uF2)上一类特殊的加性循环码的代数结构，并由此构造了一些新的二元非线性码。该项目的研究结构表明，有限环上的线性码类在自对偶码、量子纠错码以及有限域上线性码和非线性码的构造中有着重要的作用和研究价值。