Guesswork Under Linear Constraints: Exact Exponent for Coset Decoding

Abstract

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.