Non-Binary LDPC Code Design for the Poisson PPM Channel

This paper investigates the design of non-binary protograph low-density parity-check codes for the Poisson channel with <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>-ary pulse position modulation. The field order over which the code is constructed is matched to the pulse position modulation order yielding a coded modulation scheme. The optimization of the low-density parity-check code structure is performed via protograph density evolution on a surrogate <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>-ary erasure channel. The surrogate design is illustrated to be not only accurate, but also <italic>robust</italic> for a range of practical values of channel background noise and various modulation orders. As a result the proposed codes show excellent performance over the Poisson channel with pulse position modulation outperforming competing schemes. As a side-product of this paper, finite-length benchmarks on the block error probability are provided, together with a union bound to characterize the code performance in the error floor region.