五元域上LCD码的构造

On the Construction of LCD Codes over F_5

基于最优线性码与射影几何理论,针对不同码长最优码的距离特性,研究了低维五元最优LCD码的构造。首先利用删截等方法构造了较小码长的三维和四维最优线性码以及最优LCD码;其次,借助部分已知矩阵和删截等方法构造了较大码长的三维和四维最优线性码以及最优LCD码;最后,利用已知最优LCD码和特殊码长最优自正交码构造了任意大码长的最优LCD码,完全解决了三维和四维最优LCD码的构造问题。这些LCD码的构造方法对于五元高维最优LCD码以及一般域上最优LCD码的研究具有重要的理论指导意义。

Focuses on the construction of low-dimensional optimal LCD codes over F5 on the basis of the theory of optimal codes, projective geometry in view of the characteristic of the distance of optimal codes. First, via some known and puncturing, the paper constructs a short-distance optimal and optimal LCD codes of dimension 3 and 4. Then the theory of projective geometry is used to construct those codes with long-distance. Moreover, with the combination of the known optimal LCD codes and optimal self-orthogonal codes, the paper can construct any long-distance optimal LCD codes. With these methods, all the optimal LCD codes of dimension 3 and 4 can be solved. The theories and methods of constructing LCD codes have a certain of important guiding significance for the study of high-dimensional optimal LCD codes over F5 and those over general field.