AI 中文总结
本文建立了随机二元线性码在陪集解码中约束猜谜的ρ阶矩的精确指数增长率，并推广到任意权重枚举器、列表猜谜、二阶精炼以及通用性定理，应用于q元扩展和规则LDPC码。
AI 中文摘要
我们建立了约束猜谜 $G_{\mathrm{coset}}$ 的 $\rho$ 阶矩的精确指数增长率——在 i.i.d.\\ Bernoulli$(p)$ 噪声下，随机二元线性码的陪集中真实噪声向量的秩：\\( \lim_{n\to\infty} \frac{1}{n}\log_2\Eb\\!\left[G_{\mathrm{coset}}^{\rho}\right] = \rho\\,h_{\frac{1}{1+\rho}}(p)\\;+\\;\rho(R-1), \\, \rho>0, \\) 其中 $h_\alpha(p)$ 是二元Rényi熵，$R=k/n$ 是码率。该指数相对于无约束的Arıkan--Merhav指数恰好向下偏移 $\rho(1-R)$，每个 $n(1-R)$ 个奇偶校验贡献相等。有限长仿真确认了从下方收敛。我们进一步建立了：(i)~一个传递定理，将配分函数指数表示为任意权重枚举器增长率 $g(\delta)$ 的函数；(ii)~$L_n$-列表（“第 $k$ 个”）约束猜谜的精确指数；(iii)~一个尖锐的二阶精炼，阶数为 $\rho\log_2 n$。在二元 i.i.d.\\ 设置之外，我们证明了一个通用性定理：对于任何权重枚举器以速率 $g_{\mathcal{E}}(\delta)$ 集中的码系综 $\mathcal{E}$，猜谜指数等于 $(1+\rho)\psi_{1/(1+\rho)}(g_{\mathcal{E}})-\rho\\,\psi_1(g_{\mathcal{E}})$，其中 $\psi_\alpha(g)=\sup_\delta[g(\delta)+\alpha\ell(\delta)]$。作为具体应用，我们将该定理实例化到 $q$ 元扩展，$\Lambda_q(\rho)=\rho\\,h^{(q)}_{1/(1+\rho)}(P)+\rho(R-1)\log_2 q$，以及Gallager规则LDPC系综，通过系综平均权重枚举器的精确有限长恒等式获得了闭式猜谜指数。
英文摘要
We establish the exact exponential growth rate of the $ρ$-th moment of the constrained guesswork $G_{\mathrm{coset}}$ -- the rank of the true noise vector within its syndrome coset of a random binary linear code under i.i.d.\ Bernoulli$(p)$ noise: \( \lim_{n\to\infty} \frac{1}{n}\log_2\Eb\!\left[G_{\mathrm{coset}}^ρ\right] = ρ\,h_{\frac{1}{1+ρ}}(p)\;+\;ρ(R-1), \, ρ>0, \) where $h_α(p)$ is the binary Rényi entropy and $R=k/n$ is the code rate. The exponent shifts down by exactly $ρ(1-R)$ relative to the unconstrained Arıkan--Merhav exponent, with each of the $n(1-R)$ parity checks contributing equally. Finite-length simulations confirm convergence from below. We further establish: (i)~a transfer theorem expressing the partition-function exponent in terms of an arbitrary weight-enumerator growth rate $g(δ)$; (ii)~the exact exponent for $L_n$-list (``$k$-th'') constrained guesswork; and (iii)~a sharp second-order refinement of order $ρ\log_2 n$. Beyond the binary i.i.d.\ setting, we prove a universality theorem: for any code ensemble $\mathcal{E}$ whose weight enumerator concentrates at rate $g_{\mathcal{E}}(δ)$, the guesswork exponent equals $(1+ρ)ψ_{1/(1+ρ)}(g_{\mathcal{E}})-ρ\,ψ_1(g_{\mathcal{E}})$, where $ψ_α(g)=\sup_δ[g(δ)+α\ell(δ)]$. As concrete applications, we instantiate this theorem for the $q$-ary extension, $Λ_q(ρ)=ρ\,h^{(q)}_{1/(1+ρ)}(P)+ρ(R-1)\log_2 q$, and for Gallager's regular LDPC ensemble, obtaining a closed-form guesswork exponent via an exact finite-length identity for the ensemble-average weight enumerator.
