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 will consider simplicial complexes with one maximal element. The relation of the weights of codewords in two special codes obtained from simplicial complexes is illustrated by using LFSR sequences. And then we determine the parameters of these codes with the help o...
