The general procedure for determing a parity check matrix given n, k, q, and d (not from generator matrix)

Question

Is there a generic procedure for finding the parity check matrix in a linear code of $[n, k]$ over a finite field $F_q$ with a minimum weight of $d$? I am given $n$, $k$, $q$, and $d$, but I don't know the procedure for finding the parity check matrix from that. I understand how to determine the parity check matrix from the generator matrix but I'm not too sure about how to determine the parity check matrix directly from n, k, q, and d.

Answer 1

Recall that a parity check matrix of a linear code $C\subseteq \mathbb{F}^n$ is simply a generator matrix for the dual code

$$C^\perp=\{x\in \mathbb{F}^n \,|\, \forall y\in C : \langle x, y\rangle =0\},$$

where

$$\langle \cdot,\cdot\rangle : \mathbb{F}^n\times \mathbb{F}^n \to \mathbb{F},$$ $$\bigl((x_1,\dots,x_n),(y_1,\dots,y_n)\bigr)\mapsto \sum_{j=1}^n x_jy_j.$$

So what you can do is find a basis of $C^\perp$ and from that construct a generator matrix for $C^\perp$, which in turn will be a parity check matrix for $C$.