Some Short-Length Girth-8 QC-LDPC Codes From Primes of the Form t2 + 1

We propose a simple algebraic construction for girth-8 regular QC-LDPC codes of short lengths, a few hundreds, based on the square matrix from some prime integers of the form
$t^{2}+1=P$
and a multiplication table method. We generalize the conventional multiplication table method in a way that the size
$T$
of the circular permutation matrix (CPM) can be different from the modulus
$M$
in the calculation of the exponent matrix. We classify and suggest the parameters
$P\leq M\leq T$
with
$M=kP$
so that the resulting codes have girth 8. In particular, we prove the existence of a threshold
$T_{0}$
so that the resulting code will always have girth 8 if
$T > T_{0}$
is used, given that
$M=kP$
. Finally, we present various simulation results and theoretical analysis, one of which shows that the proposed codes of length around 250 have an additional coding gain of about 0.4 dB over the 5G NR LDPC codes.