On the minimum weight of simple full-length array LDPC codes

We investigate the minimum weights of simple full-length array LDPC codes (SFA-LDPC codes). The SFA-LDPC codes are a subclass of LDPC codes, and constructed algebraically according to two integer parameters p and j. Mittelholzer and Yang et al. have studied the minimum weights of SFA-LDPC codes, but the exact minimum weights of the codes are not known except for some small p and j. In this paper, we show that the minimum weights of the SFA-LDPC codes with j = 4 and j = 5 are upper-bounded by 10 and 12, respectively, independent from the prime number p. By combining the results with Yang's lower-bound limits, we can conclude that the minimum weights of the SFA-LDPC codes with j = 4 and p > 7 are exactly 10 and those of the SFA-LDPC codes with j = 5 are 10 or 12.