Generator and parity-check matrices over a finite field

Let $C\le F^n$ have dimension $k$. A generator matrix for $C$ is a matrix $G\in M_{n\times k}(F)$ with linearly independent columns and column space $C$. A parity-check matrix for $C$ is a matrix $H\in M_{(n-k)\times n}(F)$ with linearly independent rows and nullspace $C$. With column-vector conventions, $G:F^k\to F^n$ and $H:F^n\to F^{n-k}$; the product $HG$ is defined and is the zero matrix. Existence for every code, the identity $HG=0$, and the boundary shapes are proved in Every finite-field linear code has generator and parity-check matrices.

The row vectors of $H$, identified with their coordinate columns by transpose and the bilinear dot product, span $C^{\perp }$. No Hermitian conjugation is used. The matrices have shapes $n\times k$ and $(n-k)\times n$, including $n\times 0$ and $0\times n$ matrices.