In this article we mainly study linear codes over $\mathbb{F}_{2^n}$ and their binary subfield codes. We construct linear codes over $\mathbb{F}_{2^n}$ whose defining sets are the certain subsets of $\mathbb{F}_{2^n}^m$ obtained from mathematical objects called simplicial complexes. We use a result in LFSR sequences to illustrate the relation of the weights of codewords in two special codes obtained from simplical complexes and then determin the parameters of these codes. We construct fiveinfinite families of distance optimal codes and give sufficient conditions for these codes to be minimal.

翻译：本文主要研究$\mathbb{F}_{2^n}$上的线性码及其二元子域码。我们构造了定义集为$\mathbb{F}_{2^n}^m$中特定子集（这些子集来源于称为单纯复形的数学对象）的$\mathbb{F}_{2^n}$上线性码。利用线性反馈移位寄存器序列中的结论，阐明了从单纯复形获得的两类特殊码中码字权重的关联关系，进而确定了这些码的参数。我们构造了五类无穷族距离最优码，并给出了这些码为极小码的充分条件。