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Error Exponents of Typical Random Codes

机译：典型随机码的误差指数

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摘要

We define the error exponent of the typical random code (TRC) as the long-block limit of the negative normalizednexpectation of the logarithmnof the error probability of the random code, as opposed to the traditional random coding error exponent, which is the limit of the negative normalizednlogarithm of the expectationnof the error probability. For the ensemble of uniformly randomly drawn fixed composition codes, we provide exact error exponents of TRCs for a general discrete memoryless channel and a wide class of (stochastic) decoders, collectively referred to as the generalized likelihood decoder (GLD). This ensemble of fixed composition codes is shown to be no worse than any other ensemble of independent codewords that are drawn under a permutation-invariant distribution (e.g., independent identically distributed codewords). We also present relationships between the error exponent of the TRC and the ordinary random coding error exponent, as well as the expurgated exponent for the GLD. Finally, we demonstrate that our analysis technique is applicable also to more general communication scenarios, such as list decoding (for fixed-size lists) as well as the decoding with an erasure/list option in Forney’s sense.
机译：我们将典型随机码（TRC）的错误指数定义为负归一化n <斜体xmlns：mml =“ http://www.w3.org/1998/Math/MathML” xmlns：xlink = “ http://www.w3.org/1999/xlink“>对随机码错误概率的对数期望值（对数），与传统的随机编码错误指数相反，后者是负数的极限normalizedn <斜体xmlns：mml =“ http://www.w3.org/1998/Math/MathML” xmlns：xlink =“ http://www.w3.org/1999/xlink”>期望的对数无错误概率。对于均匀随机绘制的固定合成码的集合，我们为一般的离散无记忆通道和各种（随机）解码器（统称为广义似然解码器（GLD））提供了TRC的精确误差指数。示出了这种固定组成码的集合不比在排列不变分布下绘制的任何其他独立码字的集合（例如，独立的相同分布的码字）更差。我们还介绍了TRC的错误指数与普通随机编码错误指数之间的关系，以及GLD的expexped指数。最后，我们证明了我们的分析技术也适用于更一般的通信场景，例如列表解码（用于固定大小的列表）以及福尼意义上带有“擦除/列表”选项的解码。

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