I am looking for a way to construct an LDPC (Low-Density Parity-Check) Code $C$ of length $n$ and minimum distance $d_C$ that scales linearly to $n$, meaning $d_C = \delta n$ for $\delta \in (0,1)$.
I have been looking through the state-of-the-art papers about the construction of LDPC Codes, and seeing their minimum distance. Some codes are quasi-cylic (QC-LDPC) and cylic (C-LDPC), and are supposed to have "good" minimum distance, but they only scale to $\sqrt n$ at most.
Some papers efficiently construct such codes, but with the case over $GF(q)$ where $q = 2^m$, like Tanner Codes and Distance amplification for example.
Some other papers state, or even try to prove theoretically that such codes with $d_C = \delta n$ can be found, but dont' provide any partical way for the construction (for example), and don't reference any other papers for such constructions.
So I ask this question after a lot of research : is there any efficient algorithm to construct LDPC codes $C$ of length $n$ and minimum distance that scales linearly to the length $d_C = \delta n$ (over any $GF(q)$)