Say we know the definition of a general linear code over a prime field. It could be something like this for example:
$$C = \{ x_1x_2\ldots x_n | x_i \in \mathbb{F}_q, \sum_{i=1}^n x_i^q \equiv 0 \pmod{q}\} $$
Is there a way to derive its parity check matrix from this information?
In your specific example, since $x^q=x,$ for all $x\in F_q,$ your defining equation is equivalent to $$\sum_{i=1}^n x_i\equiv 0 \mod q$$
This defines a linear code.
More generally applying a Groebner basis algorithm to the defining equations will give you a generating set.