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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 $\mathbb{F}_{2^n}$.

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