Guesswork Under Linear Constraints: Exact Exponent for Coset Decoding

2026-06-30

AI Summary

This work investigates the exponential growth rate of the ρ-th moment of the number of guesses required to recover the true noise vector in syndrome decoding of random binary linear codes under i.i.d. Bernoulli noise. By integrating Rényi entropy, weight enumerating functions, large deviation theory, and finite-length identities, the authors derive—for the first time—an exact asymptotic exponent for guessing complexity under linear constraints: ρ h_{1/(1+ρ)}(p) + ρ(R−1). This expression reveals that each parity-check equation contributes equally to the exponent. The result holds for any ensemble of codes defined via a weight enumerating function and extends naturally to multi-list, q-ary, and Gallager-regular LDPC code settings. Numerical simulations confirm that the asymptotic exponent provides a lower bound that is approached from below at finite block lengths.

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.

Problem

• guesswork
• linear constraints
• coset decoding
• Rényi entropy
• weight enumerator

Innovation

• constrained guesswork
• exact exponent
• Rényi entropy
• weight enumerator
• universality theorem