F4上二维最优自正交码的分类

根据四元自正交码的重量特点,研究二维最优自正交码的生成矩阵与重量分布之间的关系.通过引入二维四元码的定义向量和射影重量概念,利用Simplex码的码字构成的矩阵,建立二维最优自正交码的存在性与整数方程组的非负解之间的联系,将确定二维最优正交码的生成矩阵问题转化为求解整数方程组的非负解.对于给定码长,首先由Griesmer界确定二维最优自正交码的距离;然后,通过求解整数方程组的非负解,确定出所有二维最优自正交码的生成矩阵和重量多项式;依据二维最优自正交码的生成矩阵,利用矩阵的初等行变化、向量的坐标置换和元素的共轭变换,判断二维最优自正交码的等价性;最后,完全解决了二维最优自正交码的分类问题,给出互不等价的二维最优自正交码的生成矩阵与重量多项式.

Classification of Two Dimensional Optimal Self-orthogonal Codes over F4

According to the weight character of quaternary self-orthogonal codes, the relations between two dimensional optimal self-orthogonal codes and their weight distribution are discussed. By introducing two definitions of defying vector and projective weight vector of quaternary linear codes and using the matrix constructed with simplex codes, the relations between two dimensional optimal self-orthogonal codes and the non negative integral solutions of equation systems are setup. And, the existence problem of two dimensional optimal self-orthogonal codes is changed into the problem of determining the non negative integral solutions of equation systems.For the given code length, firstly, the minimum distance of two dimensional optimal self-orthogonal codes is determined by Griesmer bound, then, the generator matrices and the weight polynomials of all optimal self-orthogonal codes of this length are determined through solving integral equation systems. According to generator matrices obtained, equivalent relations among these optimal self-orthogonal codes are discussed by using elementary row transformations of matrices, coordinates permutations of vectors and conjugate transformations of elements. Finally the classification of two dimensional optimal self-orthogonal codes is solved, and the generator matrices of these non equivalent two dimensional optimal self-orthogonal codes and their weight polynomials are also presented. polynomials are also presented.