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A coding scheme for transmission of a bit maps a given bit to a sequence ofchannel inputs (called the codeword associated to the transmitted bit). In thispaper, we study the problem of designing the best code for a discrete Poissonchannel with memory (under peak-power and total-power constraints). The outputsof a discrete Poisson channel with memory are Poisson distributed randomvariables with a mean comprising of a fixed additive noise and a linearcombination of past input symbols. Assuming a maximum-likelihood (ML) decoder,we search for a codebook that has the smallest possible error probability. Thisproblem is challenging because error probability of a code does not have aclosed-form analytical expression. For the case of having only a total-powerconstraint, the optimal code structure is obtained, provided that theblocklength is greater than the memory length of the channel. For the case ofhaving only a peak-power constraint, the optimal code is derived for arbitrarymemory and blocklength in the high-power regime. For the case of having boththe peak-power and total-power constraints, the optimal code is derived formemoryless Poisson channels when both the total-power and the peak-power boundsare large.

Transmission of a Bit over a Discrete Poisson Channel with Memory