Coding theory exercises using MAGMA, I

Let $\mathbb{F}$ be a finite field. Consider a short exact sequence of vector spaces

$$0 \rightarrow \mathbb{F}^k \stackrel{\gamma}{\rightarrow} \mathbb{F}^n \stackrel{\theta}{\rightarrow} \mathbb{F}^{n-k} \rightarrow 0.$$

A linear code is the image of $\gamma$ . Since the sequence is exact, a vector $v\in \mathbb{F}^n$ is a codeword if and only if $\theta(v)=0$ . If $\mathbb{F}^i$ is given the usual standard vector space basis then the matrix of $\gamma$ is the generating matrix and the matrix of $\theta$ is the parity check matrix.