有限环上的常循环码及其应用研究

常循环码具有丰富的代数结构，其编码与译码电路容易实现，在实践中被广泛应用。本项目拟研究有限环上常循环码及其在构造量子纠错码中的应用。首先，利用环同构给出有限环上任意长度的常循环码的代数结构，并结合有限交换环的理想理论分类某些具体有限环上的常循环码。其次，利用离散的傅里叶变换，确定有限环上常循环自正交码和常循环自对偶码的存在条件，给出常循环自对偶码的计数，并分类某些有限环上常循环自正交码和常循环自对偶码。最后，将有限环特别是多项式剩余类环上常循环自正交码应用于构造参数较好的量子纠错码。本项目的研究丰富了有限环上常循环码理论，拓宽了量子纠错码构造的途径，为实现数字通信和量子通信的可靠传输提供理论保障。

Constacyclic codes have a rich algebraic structure, and their encoding and decoding circuits can be carried out easily. Thus, constacyclic codes have been widely used in practice. This project studies constacyclic codes over finite rings and their application to the construction of quantum error-correcting codes. Firstly, we establish the algebraic structure of constacyclic codes of arbitrary lengths over finite rings using ring isomorphism. With the ideal theory of finite commutative rings, constacyclic codes over some finite rings are classified and enumerated. Secondly, we give the existence conditions of constacyclic self-orthogonal and self-dual codes using discrete Fourier transform. The enumeration of constacyclic self-dual codes under the existence conditions is given, and constacyclic self-orthogonal codes and self-dual codes over some finite rings are constructed. Finally, constacyclic self-orthogonal codes over finite rings, especially over polynomial residue rings are applied to the construction of quantum error-correcting codes with good parameters. The project enriches the theory of constacyclic codes over finite rings, and broadens the approach to construct quantum error-correcting codes. It also provides a theoretic guarantee for the reliability transmission of digital communication and quantum communication.