arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2007.02330cs.ITcs.CCmath.IT
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具有一般失真能力信道的共享随机模型中的通用码

Universal codes in the shared-randomness model for channels with general distortion capabilities

Bruno Bauwens, Marius Zimand

摘要:

提出了新的通用信道编码模型。与为特定类型的信道设计的标准代码不同,我们最通用的通用代码使通信在每个信道上都具有弹性,只要噪声水平低于容许范围,其中信道的噪声水平t是其模糊性的对数(可以扭曲成给定的字符串的最大数量)。其他更受限制的通用码仍然适用于大类的自然信道。在通用代码中,编码是独立于信道的,但解码函数知道信道的类型。我们允许编码和解码功能共享随机性,这是不可用的通道。信道可以执行的攻击类型有两种情况。在不经意场景中,码字属于加法组,并且信道通过添加来自固定集合的向量来使码字失真。该选择基于消息和编码函数,而不是码字。在汉明场景中,信道知道码字并且是完全对抗的。对于通用码,有两个感兴趣的参数:速率,它是消息长度k和码字长度n之间的比率,以及共享随机比特的数量。我们证明了在这两种情况下都存在速率为1-t/n-o(1)的通用码,这是对o(1)项取模的最优码。共享的随机位的数量是O(log n)在不经意的情况下,和O(n)在汉明的情况下,其中,对于典型的值的噪声水平,我们证明是最佳的,模隐藏在O()符号中的常数。在这两种情况下,通用编码是在n中的时间多项式中完成的,但是信道相关的解码过程通常不是有效的。对于某些较弱的信道类,我们构造了具有多项式时间编码和译码的通用码。

英文摘要:

We put forth new models for universal channel coding. Unlike standard codes which are designed for a specific type of channel, our most general universal code makes communication resilient on every channel, provided the noise level is below the tolerated bound, where the noise level t of a channel is the logarithm of its ambiguity (the maximum number of strings that can be distorted into a given one). The other more restricted universal codes still work for large classes of natural channels. In a universal code, encoding is channel-independent, but the decoding function knows the type of channel. We allow the encoding and the decoding functions to share randomness, which is unavailable to the channel. There are two scenarios for the type of attack that a channel can perform. In the oblivious scenario, codewords belong to an additive group and the channel distorts a codeword by adding a vector from a fixed set. The selection is based on the message and the encoding function, but not on the codeword. In the Hamming scenario, the channel knows the codeword and is fully adversarial. For a universal code, there are two parameters of interest: the rate, which is the ratio between the message length k and the codeword length n, and the number of shared random bits. We show the existence in both scenarios of universal codes with rate 1-t/n - o(1), which is optimal modulo the o(1) term. The number of shared random bits is O(log n) in the oblivious scenario, and O(n) in the Hamming scenario, which, for typical values of the noise level, we show to be optimal, modulo the constant hidden in the O() notation. In both scenarios, the universal encoding is done in time polynomial in n, but the channel-dependent decoding procedures are in general not efficient. For some weaker classes of channels we construct universal codes with polynomial-time encoding and decoding.

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