Asymptotic estimation of the fraction of errors correctable by $q$-ary LDPC codes

We consider an ensemble of random $q$-ary LDPC codes. As constituent codes, we use $q$-ary single-parity-check codes with $d=2$ and Reed–Solomon codes with $d=3$. We propose a hard-decision iterative decoding algorithm with the number of iterations of the order of the logarithm of the code length. We show that under this decoding algorithm there are codes in the ensemble with the number of correctable errors linearly growing with the code length. We weaken a condition on the vertex expansion of the Tanner graph corresponding to the code.