A coding scheme for transmission of a bit maps a given bit to a sequence of channel inputs (called the codeword associated with the transmitted bit). In this paper, we study the problem of designing the best code for a discrete Poisson channel with memory (under peak-power and total-power constraints). The outputs of a discrete Poisson channel with memory are Poisson distributed random variables with a mean comprising of a fixed additive noise and a linear combination of past input symbols. Assuming a maximum-likelihood (ML) decoder, we search for a codebook that has the smallest possible error probability. This problem is challenging because error probability of a code does not have a closed-form analytical expression. For the case of having only a total-power constraint, the optimal code structure is obtained, provided that the blocklength is greater than the memory length of the channel. For the case of having only a peak-power constraint, the optimal code is derived for arbitrary memory and blocklength in the high-power regime. For the case of having both the peak-power and total-power constraints, the optimal code is derived for memoryless Poisson channels when both the total-power and the peak-power bounds are large.

中文翻译：

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通过带记忆的离散泊松通道传输比特

用于传输位的编码方案将给定位映射到通道输入序列（称为与传输位相关联的码字）。在本文中，我们研究了为具有内存的离散泊松通道设计最佳代码的问题（在峰值功率和总功率约束下）。带记忆的离散泊松通道的输出是泊松分布的随机变量，其均值由固定的加性噪声​​和过去输入符号的线性组合组成。假设最大似然（ML）解码器，我们搜索具有最小可能错误概率的码本。这个问题具有挑战性，因为代码的错误概率没有封闭形式的解析表达式。对于只有总功率约束的情况，得到最优代码结构，假设块长度大于通道的内存长度。对于只有峰值功率约束的情况，在高功率状态下为任意存储器和块长度推导出最优代码。对于同时具有峰值功率和总功率约束的情况，当总功率和峰值功率界限都很大时，为无记忆泊松信道推导出最优代码。