arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1004.4017cs.ITcs.CCmath.IT
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用于计算简单信道的最佳速率码构造

Optimal-Rate Code Constructions for Computationally Simple Channels

Venkatesan Guruswami, Adam Smith

摘要:

我们考虑计算有界信道的编码方案,只要(a)误差的分数以高概率由参数$p$界定,并且(b)添加误差的过程可以由足够简单的电路描述,该编码方案就可以引入任意的误差集。这样的信道模型的代码是有吸引力的,因为像标准对抗性错误的代码,他们可以处理信道的真实行为是未知的或随着时间的推移而变化。对于两类通道,我们提供了明确的,有效的编码/解码的最佳速率,只有低效解码的代码是以前已知的代码。在每种情况下,我们提供一个编码器/解码器,适用于类中的每个通道。编码器是随机的,并且概率被编码器的(本地的,解码器未知的)硬币和通道的硬币所接受。(1)唯一的解码加法错误:我们给出了第一个构造多项式时间的可编码/可解码的代码加法(a.k.a.不经意的)信道,其实现香农容量$1-H(p)$。这些信道将权重至多为$pN$的任意误差向量$e\in\{0,1\}^N$添加到发送的字;向量$e$可以取决于代码,但不取决于特定的发送字。(2)列表解码的多项式时间的信道:对于每个恒定的$c>0$,我们给出了一个Monte Carlo构造的代码与最佳速率(任意接近$1-H(p)$),有效地恢复一个简短的列表包含正确的消息,具有高概率的电路的大小最多$N^c$描述的信道。我们证明放松列表解码显示,即使有界的渠道,唯一可解码的代码不能有积极的$p>1/4$率。

英文摘要:

We consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter $p$ and (b) the process which adds the errors can be described by a sufficiently simple circuit. Codes for such channel models are attractive since, like codes for standard adversarial errors, they can handle channels whose true behavior is unknown or varying over time. For two classes of channels, we provide explicit, efficiently encodable/decodable codes of optimal rate where only inefficiently decodable codes were previously known. In each case, we provide one encoder/decoder that works for every channel in the class. The encoders are randomized, and probabilities are taken over the (local, unknown to the decoder) coins of the encoder and those of the channel. (1) Unique decoding for additive errors: We give the first construction of a polynomial-time encodable/decodable code for additive (a.k.a. oblivious) channels that achieve the Shannon capacity $1-H(p)$. These channels add an arbitrary error vector $e\in\{0,1\}^N$ of weight at most $pN$ to the transmitted word; the vector $e$ can depend on the code but not on the particular transmitted word. (2) List-decoding for polynomial-time channels: For every constant $c>0$, we give a Monte Carlo construction of an code with optimal rate (arbitrarily close to $1-H(p)$) that efficiently recovers a short list containing the correct message with high probability for channels describable by circuits of size at most $N^c$. We justify the relaxation to list-decoding by showing that even with bounded channels, uniquely decodable codes cannot have positive rate for $p>1/4$.

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