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.

Value taken for terms from general theory of linear codes

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