几类码的结构和距离及其应用研究

Constacyclic codes have rich algebraic structures and have been widely used in practical communications. Cyclic permutation codes are derived from cyclic codes. They can be used to construct symbol-pair codes, quantum codes, optimal frequency-hopping sequences and to design cryptography etc. Matrix product code is a new code utilizing matrix and some classical codes. Galois inner product is a generalization of Euclidean inner product and Hermitian inner product. The hull of a linear code is the intersection of itself and its dual, and its’ structure has important applications in computing automorphism groups of codes, studying equivalence of codes and designing good decoding algorithms. ..The project is scheduled to study four problems of the interrelated objects mentioned above. Firstly, we shall study the algebraic structures and minimal Hamming distances of constacyclic codes of arbitrary different lengths over Galois rings, and generalize the obtained results to finite chain rings. Under this general frame, we shall characterize precisely the structures of constacyclic codes over finite fields; we also study their weight distributions, the relationship between minimal Hamming distances and minimal symbol-pair distances. The constructions of MDS symbol-pair codes will also be considered. Secondly, we will focus on the study of cyclic permutation codes; we are specially interested in the aspects of enumerations, structures of cyclic permutation codes, and their applications on constructions of optimal frequency-hopping sequences. Thirdly, we try to characterize some lower bound on homogeneous distance of matrix product codes over a commutative principal ideal ring. Finally, by using Galois inner product, the hulls of linear codes will be discussed, and we shall apply the obtained general results to characterize the hulls of constacyclic codes and matrix product codes.

常循环码结构丰富且应用广泛，循环置换码是由循环码导出的码，它们可以构造symbol-pair码、量子码、最优跳频序列及应用于密码设计。矩阵积码是利用矩阵和一些码构造出的新码。Galois内积是欧氏内积和Hermitian内积的推广。线性码的hull是码与其对偶码的交，在计算码的自同构群、刻画码等价及设计好的译码算法等方面有重要应用。..本项目拟对上述相互联系的对象研究四个问题。一是研究Galois环上各种长度常循环码的结构和距离，并推广到有限链环；在这个一般框架下研究有限域上码的Hamming距离和symbol-pair距离的关系及MDS symbol-pair码的构造。二是研究循环置换码的计数、结构，并应用其构造新的最优跳频序列。三是研究任意有限交换主理想环上矩阵积码齐次距离下界的一般性刻画。四是研究Galois内积下有限域上线性码的hull，并用于刻画常循环码和矩阵积码的hull。

本项目利用代数、有限域和有限环理论以及群表示论等研究了几类码在不同内积下的代数结构和距离及其应用。具体成果如下：(1) 我们在有限环上引入了$\sigma$-内积，并刻画了有限链环$F_{p^m}+uF_{p^m}, u^2=0$上长度为素数$p$的幂的常循环码及其$\sigma$-对偶码的结构；我们将有限域上的常循环码推广到有限环上的不可逆常循环码，刻画了有限交换主理想环上不可逆常循环码及其对偶码的结构；我们构造了一类有限链环上具有较少Lee-重量的码的无限类，证明了在特定条件下这类码在Gray映射下是达到Griesmer界的二重量码；我们获得了有限域上特定长度的重根常循环码的Hamming距离和$b$-symbol距离及MDS symbol-pair码的构造；我们还确定了有限域上几类特殊的BCH码的维数和Bose距离。(2) 我们完整地给出了半单情形下循环置换码的计数公式和构造方法，主要的创新是将挑选码字个数最多的循环置换码的问题归结为有限阿贝尔群与其一个特定子群的陪集代表元的选取问题。(3) 我们得到了有限交换主理想环上齐次距离是度量的必要充分条件，由此我们获得了任意有限交换主理想环上矩阵积码齐次距离下界的一般性刻画。(4) 我们给出了线性码在置换等价下其Galois hull的维数的刻画及计算码的hull的维数的方法，证明了当域的基数$q>4$时，存在Galois LCD码；我们给出了矩阵积码和RS码的hull的结构和维数的刻画。(5) 我们构造了新的自对偶MDS码；借助Sidon空间，我们给出了循环常维数子空间码的几类新构造，在部分情况下对循环常维数子空间码的猜想给出了证明；我们还构造出了有限域上新的具有较少重量的线性码等。