# Underlying codes in 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 $$\mathbb{F}_{2^n}$$.