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Ratio List Decoding

机译：比率列表解码

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摘要

We extend the notion of list decoding to ratio list decoding which involves a list decoder whose list size is specified as a function of the number of messages M-n and the block length n. We present the necessary and sufficient conditions on Mn for the existence of code sequences which enable reliable list decoding with respect to the desired list size L(M-n, n). It is shown that the ratio-capacity, defined as the supremum of achievable normalized logarithms of the ratio r(M-n, n) = M-n/L(M-n, n), is equal to the Shannon channel capacity C, for both stochastic and deterministic encoding. Allowing for random list size, we are able to deduce some properties of identification codes, where the decoder's output can be viewed as a list of messages corresponding to decision regions that include the channel output. We further address the regime of mismatched list decoding, in which the list constitutes of the codewords that accumulate the highest score values (jointly with the channel output) according to some given function. We study the case of deterministic encoding and mismatched ratio list decoding. We establish similar necessary and sufficient conditions for the existence of code sequences which enable reliable mismatched list decoding with respect to the desired list size L(M-n, n), and we show that the ratio-capacity with mismatched decoding is equal to the mismatch capacity. Focusing on the case of an exponential list size L-n = e(n Theta), its comparison with ordinary mismatched decoding shows that the increase in capacity is by Theta bits per channel used for all channels and decoding metrics. Several properties of the average error probability in the setup of mismatched list decoding with deterministic list size are provided.
机译：我们将列表解码的概念扩展为比率列表解码，它涉及一个列表解码器，其列表大小根据消息数M-n和块长度n指定。对于存在的代码序列，我们给出了Mn上的必要条件和充分条件，这些代码序列使得能够针对所需列表大小L（M-n，n）进行可靠的列表解码。结果表明，对于随机和确定性，比率容量定义为比率r（Mn，n）= Mn / L（Mn，n）的可归一化对数的和，等于香农通道容量C编码。考虑到随机列表的大小，我们可以推断出识别码的某些属性，在这些属性中，解码器的输出可以看作是与包含信道输出的决策区域相对应的消息列表。我们进一步解决不匹配列表解码的问题，其中列表由根据某些给定功能累积最高得分值（与通道输出一起）的代码字组成。我们研究确定性编码和不匹配比率列表解码的情况。我们为代码序列的存在建立了类似的必要条件和充分条件，这些条件使得能够对所需列表大小L（Mn，n）进行可靠的不匹配列表解码，并且我们证明了不匹配解码的比率容量等于不匹配容量。着眼于指数列表大小L-n = e（n Theta）的情况，它与普通不匹配解码的比较表明，容量的增加是通过用于所有通道和解码指标的每个通道的Theta位。提供了具有确定性列表大小的不匹配列表解码的设置中平均错误概率的几个属性。

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