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Low-density parity-check codes are commonly decoded using iterative message-passing decoders, such as the min-sum and sum-product decoders. Computer simulations demonstrate that these suboptimal decoders are capable of achieving low probability of bit error at signal-to-noise ratios close to capacity. However, current methods for analyzing the behavior of the min-sum and sum-product decoders fails to produce usable bounds on the probability of bit error. Thus, the resulting probability of bit error when using these decoders remains largely unknown for signal-to-noise ratios beyond the reach of simulation. For this reason, it is worth considering alternative methods for decoding low-density parity-check codes.
New methods for decoding low-density parity-check codes, known as finite tree-based decoders, are presented as alternative decoders for low-density parity-check codes. The goal of the finite tree-based decoders is to achieve probability of bit error comparable to that of the min-sum and sum-product decoders, while allowing for computationally tractable performance analysis. Finite tree-based decoding requires the construction of finite trees derived from the Tanner graph of the low-density parity-check code. The resulting size of the finite trees allows for current analytical techniques, such as deviation bounds and density evolution, to be used to predict the probability of bit error of finite tree-based decoding of short-to-moderate length low-density parity-check codes. Simulation results show that finite tree-based decoders are capable of outperforming current iterative decoders at high signal-to-noise ratios. Examples are also given where finite tree-based decoding provably approaches maximum-likelihood performance as the signal-to-noise ratio grows large.
A new method is also presented for lower bounding the minimum distance of low-density parity-check codes. This new lower bound is used as a cost criteria for the construction of low-density parity-check codes with both large girth and minimum-distance properties. Codes generated with this new construction technique are shown in simulations to outperform codes generated with the progressive edge-growth algorithm, using both iterative decoding and finite tree-based decoding.