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In this paper, we propose a new method based on combinatorial designs for constructing high-girth low-density parity-check (LDPC) codes. We use a 3-D lattice to generate balanced incomplete block designs based on planes and lines in the lattice. This gives families of regular LDPC codes with girths of at least 6, 8, and 10, whose parity-check matrices are all block circulant. The main advantage of this construction is that the algebraic structure leads to efficient encoders and decoders. Based on the block-circulant structure of a parity-check matrix, we present an efficient encoder that can be parallelized to improve the speed of encoding. The simulation results show that these families of LDPC codes perform very well on additive-white-Gaussian-noise channels (roughly 0.45 dB from the channel capacity) and Rayleigh fading channels (roughly 0.51 dB from the channel capacity).