  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.
