This paper deals with the Low-Density Parity-Check codes with the constituent Hamming codes (H-LDPC codes) and two different iterative hard-decision low-complexity decoding algorithms. Both algorithms are based on the same main idea: the decreasing of the syndrome weight on each step of the decoding algorithm. The first decoding algorithm uses the properties of the constituent Hamming code. The best known lower-bound on the guaranteed corrected errors fraction for the H-LDPC codes under the first decoding algorithm was obtained in 2011. The second decoding algorithm considers H-LDPC as the irregular LDPC code and uses the well-known majority decoding algorithm. The lower-bound on the guaranteed corrected errors fraction for H-LDPC code under the second decoding algorithm is introduced for the first time in this paper. Numerical results for the lower-bound, obtained in this paper for H-LDPC code under the second decoding algorithm, significantly exceed the numerical results for the best known lower-bounds, obtained previously for H-LDPC code under the first decoding algorithm.