Classical linear code over a finite field

Let $F$ be a finite field of order $q$, let $n\ge 1$, and let $C\le F^n$ be a linear subspace (Finite fields and their order, Vector space over a field, Linear subspace of a vector space). Put $k:={\dim }_{F}C$; since $F^n$ has dimension $n$, $0\le k\le n$ (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$, If ${\dim }_{F}V=n$ and $U$ is a linear subspace of $V$, then $U$ is finite-dimensional, ${\dim }_{F}U\le n$, and ${\dim }_{F}U=n$ if and only if $U=V$). A classical linear code over $F$ with a specified encoding is the data $(F,n,C,\mathrm{Enc})$, where $\mathrm{Enc}:F^k\to C$ is a linear isomorphism. Such an isomorphism exists because ${\dim }_F{F}^k=k$ and finite-dimensional vector spaces of equal dimension are isomorphic (Two finite-dimensional vector spaces over $F$ are linearly isomorphic if and only if they have the same dimension). The field is part of the data: notation $F_q$ does not identify the field with $\mathbb{Z}\mathrm{/}q\mathbb{Z}$; that quotient is a field only when $q$ is prime.

If $C=\{0\}$, then $k=0$, $F^0=\{0\}$, and the encoding is the unique map between these zero spaces. If an encoding is represented by a full-column-rank matrix $G\in M_{n\times k}(F)$, its image has dimension $k$ by A full-column-rank matrix has image dimension equal to its number of columns.

The code is the linear subspace together with a declared message encoding. No field existence or uniqueness theorem is repeated here.