This paper proposes a novel code-construction method for constructing (3, k) regular low density parity check (LDPC) codes. Other methods of designing regular LDPC codes suffer from rank-deficiency of H and high encoding complexity. The basic design of the proposed code-construction utilizes deterministic base matrices of information part and parity part in three stages of code-construction to achieve no rank-deficiency of H, no girth of 4-cycles, non-singular parity part (Hpar) and low encoding complexity. It is shown that the proposed (3, k) regular LDPC codes perform better in high code rate application. The result of (3, 42) regular code performs 0.97 dB from Shannon limit at BER 10-6 when the value of code rate R = 0.928, code length N = 14364 and encoding complexity of the order of O(1.0225 N).