# How are short exact sequences and linear codes related in coding theory?

Without going into too much detail, can some one please explain how binary linear codes and short exact sequences are related, and maybe point to some form of notes I can read on the subject?

I'm asking because apparently this is something I need to know for my coding theory exam next week. The text I'm reading doesn't even mention short exact sequences, and yet looking at old exam questions I've seen them mentioned quite frequently.

One such question asks to explain binary linear codes through short exact sequences of $\mathbb{F}_2^n$ over $\mathbb{F_2}$. How would you answer this?

Let $C$ be a $[n,k]$-linear code over the field $F$ ($= \mathbb{F}_2$) with generator matrix $G$ and parity check matrix $H$. Then the sequence
$$0 \longrightarrow F^k \stackrel{G}{\longrightarrow} F^n \stackrel{H^T}{\longrightarrow} F^{n-k} \longrightarrow 0.$$
is a short exact sequence, since $\ker H^T = C = \operatorname{im} G$.