We propose a novel construction of product codes for high-density magnetic recording based on binary low-density parity check (LDPC) codes and binary image of Reed Solomon (RS) codes. Moreover, two novel algorithms are proposed to decode the codes in the presence of both AWGN errors and scattered hard errors (SHEs). Simulation results show that at a bit error rate (bER) of approximately 10^-8, our method allows improving the error performance by approximately 1.9dB compared with that of a hard decision decoder of RS codes of the same length and code rate. For the mixed error channel including random noises and SHEs, the signal-to-noise ratio (SNR) is set at 5dB and 150 to 400 SHEs are randomly generated. The bit error performance of the proposed product code shows a significant improvement over that of equivalent random LDPC codes or serial concatenation of LDPC and RS codes.