Linear programming (LP) decoding based on the alternating direction method of multipliers (ADMM) has proved to be effective for low-density parity-check (LDPC) codes. However, for high-density parity-check (HDPC) codes, the ADMM-LP decoder encounters two problems, namely a high-density check matrix in HDPC codes and a great number of pseudocodewords in HDPC codes' fundamental polytope. The former problem makes the check polytope projection extremely complex, and the latter one leads to poor frame error rates (FER) performance. To address these issues, we introduce the even vertex algorithm (EVA) into the ADMM-LP decoding algorithm for HDPC codes, named as HDPC-EVA. HDPC-EVA can reduce the complexity of the projection process and improve the FER performance. We further enhance the proposed decoder by the automorphism groups of codes, creating diversity in the parity-check matrix. The simulation results show that the proposed decoder is capable of cutting down the average decoding time for each iteration by 30%-60%, as well as achieving near maximum likelihood (ML) performance on some BCH codes.