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.

中文翻译：

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有限向量空间的信息论

鉴于香农熵与多项式系数的增长率有关，我们表明二次熵（Tsallis 2-熵）与其$q$-形变有关；当 $q$ 是质数幂时，这些 $q$-多项式系数计算具有指定长度和维数的有限向量空间的标志。特别是，$q$-二项式系数计算给定维度的向量子空间。我们通过这种方式获得了二次熵的非可加性的组合解释，这是由标志的递归计数引起的。我们表明，其配置由标志描述的统计系统为 Tsallis 统计的最大熵原理提供了频率论的理由。然后我们引入与 $q$-二项式概率分布相关的离散时间随机过程，在时间 $n$ 生成 $\mathbb{F}_q^n$ 的向量子空间（这里 $\mathbb{F}_q$ 是 $q$ 阶的有限域）。某些“典型子空间”上的度量集中允许我们将渐近均分属性扩展到此设置。典型集合的大小由二次熵量化。当消息对应于向量空间时，我们讨论了香农理论的应用，特别是源编码。