Hamming weight and distance of a finite-field linear code

For $x=(x_0,\dots ,x_{n-1})\in F^n$, its Hamming weight is $\mathrm{wt}(x):=\mathrm{\mid }\{j<n:x_j\mathrm{\ne }0\}\mathrm{\mid }$. For $x,y\in F^n$, their Hamming distance is $d_H(x,y):=\mathrm{wt}(x-y)$. For a nonzero code $C\le F^n$, define its minimum distance by $d(C):={\min }_{0\mathrm{\ne }c\in C}\mathrm{wt}(c)$. This minimum exists because its nonempty set of values lies in $\{1,\dots ,n\}$ and the natural numbers are well ordered (The well-ordering principle). For the zero code $C=\{0\}$, set $d(C):=n+1$ by convention; this sentinel exceeds every possible Hamming weight and makes the usual radius statements vacuously valid for distinct codewords.

The code parameters are written $[n,k,d{]}_q$, where $q=\mathrm{\mid }F\mathrm{\mid }$ and $d=d(C)$. This notation records a finite-field alphabet and does not imply that $F=\mathbb{Z}\mathrm{/}q\mathbb{Z}$.

The identity $d_H(x,y)=\mathrm{wt}(x-y)$ is used over every characteristic; only in characteristic two does subtraction equal addition.