Linear encoding of a group

Definition
A linear encoding of a group $$G$$ is the following data:


 * An integer $$n$$, a field $$F_q$$ and the standard encoding for $$GL_n(F_q)$$ (as described in encoding of symmetric groups)
 * A specification of the given group $$G$$ as a subgroup of the general linear group by means of a generating set $$A$$ (preferably a small one)
 * A membership test for $$G$$ as a subgroup of the general linear group

A linear encoding can also be viewed as a concrete, or explicit, description using a failthful linear representation.

Related notions
Permutation encoding is the notion for encoding as a subgroup of a symmetric group.