 Linear code

A linear code over a finite field ${\mathbb{F}}_q$ is a linear subspace $C\subseteq {\mathbb{F}}_q^n$. Its parameters $[n,k,d{]}_q$ record word length, dimension, and minimum Hamming distance. For a nonzero code, $d$ is the least Hamming weight of a nonzero word, because the difference of two words is another word.

For example, $C=\{000,011,101,110\}$ is a binary $[3,2,2]$ code, the kernel of the parity-check matrix $(111)$. Linearity lets a generator matrix encode messages and lets the parity-check syndrome detect corruption. The zero code needs a separate convention for its minimum distance, since it has no nonzero word.