Inseon Kim

In this paper, an algebraic construction of regular QC‐LDPC codes by using the modular multiplication table mod P and Golomb rulers are proposed. It is proved that the proposed QC‐LDPC codes based on a Golomb ruler of length L have girth at least 8 if . The error performance of the proposed QC‐LDPC codes are simulated with various Golomb rulers. The proposed codes of length around 300 from the optimal 6‐mark Golomb ruler have an additional coding gain of at least 0.1 dB over 5G NR LDPC codes, 0.5 dB over those given earlier by others, both at FER 10 ⁻³ . Some non‐trivial techniques to increase the length of a given Golomb ruler with and without an additional mark for improving the performance of the codes from Golomb rulers up to 0.7 dB are also found.

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.

In this letter, we propose a new construction of girth-8 Quasi-Cyclic Low-Density Parity-Check codes (QC-LDPC) with various lengths for the global navigation satellite systems (GNSS). This scheme combines two steps. The first is the construction of a family of regular girth-8 QC-LDPC codes of various lengths and rates with two designed sequences. The second is the performance improvement of those from the first construction of half-rate cases using a proposed weight matrix so that the result becomes type-II QC-LDPC codes. This results in some final codes with short lengths of 600, 1200, and 1800, especially for future GNSS. We performed a simulation and confirmed that the proposed QC-LDPC codes of lengths 600 and 1200 have an additional coding gain about 0.3 dB at frame error rate 10 ⁻⁵ over the LDPC codes used in the Global Positioning System.