【作者】 裴君莹；

【导师】 刘三阳；

【作者基本信息】 西安电子科技大学 ， 应用数学， 2002， 博士

【摘要】 四元线性码是经典的纠错码理论在近十年间发展起采的一个重要研究方向，它与二元非线性码的构造及研究有着紧密的联系。本文讨论了四元线性码及其推广中的几个问题。 本文的主要结果为： 1．给出了判定一个四元负循环码的二元像是否是循环码的充分必要条件，并在条件成立时，给出了其二元像的结构； 2．给出了四元Goethals码的对偶码的迹表示及2-adic表示，并证明了它在Gray映射下的二元像是二元Goethals码的形式对偶码；从而可以将对非线性码二元Goethals码及其形式对偶码的研究转化为对四元线性码的研究； 3．对于类型为4^k的四元线性码，建立了联系它与其对偶码的支集重量分布之间的MacWilliams恒等式； 4．对于伽罗华环GR（4^m）上的一类循环码，以及GR（4^m）在Z₄上的一组基，给出了满足这类循环码相对于该组基展开后得到的码是四元循环码的充分必要条件，该条件简便实用； 5．建立了多项式环Z_p^e[x]中的Hensel引理及提升，证明了xⁿ-1在多项式环Z_p^e[x]中的唯一分解性，并给出了Z_p^e上所有循环码的结构； 6．将著名的Berlekamp-Massey解码算法推广到伽罗华环上，给出了伽罗华环GR(p^e，p^em)上的Berlekamp-Massey解码算法。更多还原

【Abstract】 The theory of quaternary linear codes is one of the important subjects in recent ten years, which is closely related with the study on constructions and properties of binary nonlinear codes. In this paper, several questions about quaternary codes and their generalization are discussed. Solutions presented in this thesis can be summarized as follows:- A sufficient and necessary condition is given for the negacyclic codes whose binary images under the Gray map being cyclic, and the binary images are determined when the condition is satisfied.- The trace representation and the 2-adic representation for the dual of the quaternary Goethals code are given, and its binary image is proved to be the formal dual of the binary Goethals code.- The generalized MacWilliams identities are given for the support weight distribution of a quaternary linear code with type 4k and that of its dual code.- All of the cyclic codes over the Galois ring GR(4m), and the bases for GR(4m) over Z4 are determined, such that with respect to the bases, the 4-ary image of the cyclic codes are still cyclic.- The Hensel’s Lemma and Lift over the polynomial ring Zpe [x] are discussed, and it is proved that xn - 1, where (n,p) = 1, can be uniquely factored. Furthermore, all of the cyclic codes over Zpe are determined.- The Berlekamp-Massey algorithm over the Galois ring GR(pe,.Pem) is given.更多还原