Underlying codes for Niederreiter cryptosystems Niederreiter cryptosystem is usually described by a parity check matrix $H$ over $\mathbb{F}_{2^n}$.

The minimum distance $d$ is given by

$d := min\lbrace k \text{ such that there are $k$ linearly dependant columns in $H$}\rbrace$ 

Encryption in this case is done by encoding messages $m$ of weight less than $d/2$.

Now keeping Niederreiter systems in mind, my question is, since $m$ is over $\mathbb{F}_2$, it makes sense to consider linearly dependence over $\mathbb{F}_2$ rather than $\mathbb{F}_{2^n}$. In other words, is it okay to consider code as a code over $F_{2}$ with alphabet extension or do we  have to consider it as codes over $\mathbb{F}_{2^n}$.
According to Sendrier, Niederreiter systems are only known to be secure when used with Goppa codes. All other families of codes which have been tried have been broken. You are suggesting a code superficially related to the Goppa code but probably having very different properties. Hence it is likely to be insecure (or perhaps useless for other reasons, such as low rate or low distance).

More generally, cryptosystems are surprisingly brittle. I would leave them to the experts.