Binary linear code

Definition
A binary linear code is a vector subspace $$W$$ of the power set of a finite set $$X$$. Here, the power set of a finite set is treated as a vector space via the usual XOR operation.

Message length
This is the dimension of the vector subspace.

Code length
This is the size of the set $$X$$.

Weight of the code
This is simply the minimum cardinality among all members of $$W$$ (note that each member of $$W$$ is a subset of $$X$$ so it makes sense to talk of its cardinality).

Particular cases

 * Quadratic residue code