Low-Density Parity-Check (LDPC) codes are critical for enhancing error correction in communication systems, making them a key enabler of reliable, high-speed data transmission, and a foundational technology in the development of the 6th Generation Mobile Communication Technology (6G) networks. However, traditional LDPC codes exhibit reduced error-correction efficiency in highly noisy or complex communication environments, especially at very low Signal-to-Noise Ratio (SNR). This paper proposes a low code rate scheme using LDPC-Hadamard codes, with Hadamard code constraints imposed at the check nodes. A Progressive Edge-Growth (PEG) algorithm is employed to simplify the coding process. Additionally, this paper proposes the Log-Likelihood Ratio Belief Propagation (LLR-BP) algorithm to reduce the computational complexity of the decoding process. Simulation results indicate that LDPC codes achieve a Bit Error Rate (BER) of \(10^{-5}\) at a lower SNR of \(0.3\,\text {dB}\) , in contrast to LDPC-Hadamard codes, which require an SNR of \(-0.2\,\text {dB}\) . The results demonstrate the performance improvement of LDPC-Hadamard codes in BPSK modulation is \(0.5\,\text {dB}\) superior than that of LDPC codes.