In contrast to conventional low-density parity-check (LDPC) code construction, we propose to construct a sparse decoding matrix with auxiliary variable nodes (AVNs) and then derive the corresponding high-density parity-check (HDPC) code for encoding. To leverage the sparsity gain of AVNs while controlling AVN-induced harmful substructures, we adopt a parity-check row block (PCRB) structure and show that arranging PCRBs to obtain a 4-cycle-free decoding matrix forms a class of social golfer problems (SGPs). Focusing on an explicit affine-plane-based SGP solution, we construct a family of AVN-aided affine plane (AAP) codes. For the same blocklength and minimum distance, AAP codes achieve equal or greater dimension than Reed-Muller codes and exhibit low error floors under iterative decoding, which extend finite-geometry LDPC constructions to additional code families with 4-cycle-free decoding matrices.