The Euclidean projection operation is the most complex and time-consuming of the alternating direction method of multipliers (ADMM) decoding algorithms, resulting in a large number of resources when deployed on hardware platforms. We propose a simplified line segment projection algorithm (SLSA) and present the hardware design and the quantization scheme of the SLSA. In simulation results, the proposed SLSA module has a better performance than the original algorithm with the same fixed bitwidths due to the centrosymmetric structure of SLSA. Furthermore, the proposed SLSA module with a simpler structure without hypercube projection can reduce time consuming by up to 72.2% and reduce hardware resource usage by more than 87% compared to other Euclidean projection modules in the experiments.