Guesswork Under Linear Constraints: Exact Exponent for Coset Decoding

Abstract

We establish the exact exponential growth rate of the $\rho$-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: \( \lim_{n\to\infty} \frac{1}{n}\log_2\Eb\!\left[G_{\mathrm{coset}}^{\rho}\right] = \rho\,h_{\frac{1}{1+\rho}}(p)\;+\;\rho(R-1), \, \rho>0, \) where $h_\alpha(p)$ is the binary R\'{e}nyi entropy and $R=k/n$ is the code rate. The exponent shifts down by exactly $\rho(1-R)$ relative to the unconstrained Ar{\i}kan--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 $g(\delta)$; (ii)~the exact exponent for $L_n$-list (``$k$-th'') constrained guesswork; and (iii)~a sharp second-order refinement of order $\rho\log_2 n$. Beyond the binary i.i.d.\ setting, we prove a universality theorem: for any code ensemble $\mathcal{E}$ whose weight enumerator concentrates at rate $g_{\mathcal{E}}(\delta)$, the guesswork exponent equals $(1+\rho)\psi_{1/(1+\rho)}(g_{\mathcal{E}})-\rho\,\psi_1(g_{\mathcal{E}})$, where $\psi_\alpha(g)=\sup_\delta[g(\delta)+\alpha\ell(\delta)]$. As concrete applications, we instantiate this theorem for the $q$-ary extension, $\Lambda_q(\rho)=\rho\,h^{(q)}_{1/(1+\rho)}(P)+\rho(R-1)\log_2 q$, 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.