有限向量空间信息论
Information theory with finite vector spaces
摘要：
而香农熵的增长率的多项式系数，我们表明，二次熵（Tsallis 2-熵）连接到他们的$q$ -变形;当$q$是一个素数的权力，这些$q$ -多项式系数计数标志的有限向量空间具有规定的长度和尺寸。特别地，$q$ -二项式系数计数给定维度的向量子空间。我们获得这种方式的组合解释的非可加性的二次熵，这是从递归计数的标志。我们表明，统计系统的配置所描述的旗帜提供了一个频率论的理由与Tsallis统计的最大熵原理。然后，我们介绍了一个离散时间随机过程相关联的$q$ -二项式概率分布，在时间$n$生成一个向量子空间的$\mathbb{F}_q^n$（这里$\mathbb{F}_q$是有限域的顺序$q$）。测度在某些“典型子空间”上的集中使我们能够将渐进均分性质扩展到这种设置。典型集的大小由二次熵量化。我们讨论应用香农理论，特别是信源编码，当消息对应于向量空间。
英文摘要：
Whereas Shannon entropy is related to the growth rate of multinomial coefficients, we show that the quadratic entropy (Tsallis 2-entropy) is connected to their $q$-deformation; when $q$ is a prime power, these $q$-multinomial coefficients count flags of finite vector spaces with prescribed length and dimensions. In particular, the $q$-binomial coefficients count vector subspaces of given dimension. We obtain this way a combinatorial explanation for the nonadditivity of the quadratic entropy, which arises from a recursive counting of flags. We show that statistical systems whose configurations are described by flags provide a frequentist justification for the maximum entropy principle with Tsallis statistics. We introduce then a discrete-time stochastic process associated to the $q$-binomial probability distribution, that generates at time $n$ a vector subspace of $\mathbb{F}_q^n$ (here $\mathbb{F}_q$ is the finite field of order $q$). The concentration of measure on certain "typical subspaces" allows us to extend the asymptotic equipartition property to this setting. The size of the typical set is quantified by the quadratic entropy. We discuss the applications to Shannon theory, particularly to source coding, when messages correspond to vector spaces.
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