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Low-density parity-check (LDPC) codes offer a very powerful error correction technique which allows data transmission in wireless networks at rates near the channel capacity with arbitrarily low probability of error. In this paper, we design a class of linearly encodable LDPC codes with adaptive code rates, i.e. the code rate can be adapted according to channel conditions to maximize the total capacity. Since a unique mother parity check matrix is used to construct LDPC codes with several code rates, the great advantage of the proposed codes is that a single universal encoder (decoder) is adequate to encode (decode) multi-rate codes, which makes it possible to efficiently implement multi-rate LDPC codes in a subscriber station. The implementation results into field programmable gate array (FPGA) devices indicate that a universal layered encoder for LDPC codes with 9 code rates is capable of reaching a throughput above 1.2 Gigabit per second by using 138 exclusive- OR gates and a master clock of 100 MHz. By combining multilevel modulation with the designed multi-rate LDPC codes, a transmission scheme with adaptive spectral efficiency is proposed. The simulation results indicate that the schemes with a spectral efficiency of 1, 2, 3, 4, and 5 bits/symbol/Hz can achieve extremely reliable transmission in a Rayleigh fading channel at signal-to-noise ratios (SNR) per bit of 5, 6, 10, 14, and 16 dB, respectively.