[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83459-en":3,"doc-seo-83459-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83459,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Guesswork Under Linear Constraints Exact Exponent for Coset Decoding","Establishes the exact exponential growth rate of the ρ-th moment of constrained guesswork over syndrome cosets for random binary linear codes with i.i.d. Bernoulli(p) noise. The main result gives a closed-form exponent via binary Rényi entropy and shows a downward shift by ρ(1−R) relative to the unconstrained Arıkan–Merhav exponent, with equal contributions from the n(1−R) parity checks. Finite-length simulations confirm convergence; further results include a transfer theorem, list-constrained guesswork, second-order refinement, and a universality theorem extending to code ensembles with concentrated weight-enumerator growth.","1  \narXiv :2607 .00205v 1 [ cs .IT] 30 Jun 2026  \nGuesswork Under Linear Constraints: Exact Exponent for Coset Decoding  \nHassan Tavakoli,  \nSchool of EECS,  \nOregon State University,  \n[tavakolh@oregonstate.edu](tavakolh@oregonstate.edu)  \nAbstract  \nWe establish the exact exponential growth rate of the ρ-th moment of the constrained guesswork Gcoset—the rank of the true noise vector within its syndrome coset of a random binary linear code under i.i.d. Bernoulli (p) noise: limn→∞ ~~1~~n log2 E [Gρcoset] =ρh 1  (p) + ρ (R−1), ρ > 0 , where hα (p) is the binary Rnyi entropy and R = k/n is the code rate. The exponent shifts down  \n1+ρ  \nby 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 Ln-list (“k-th”) constrained guesswork; and (iii) a sharp second-order refinement of order ρlog2 n. Beyond the binary i.i.d. setting, we prove a universality theorem: for any code ensemble E whose weight enumerator concentrates at rate gE (δ), the guesswork exponent equals (1+ρ)ψ 1/(1+ρ)(gE )−ρψ1 (gE ), where ψα (g) = supδ [g(δ)+αℓ(δ)] . As concrete applications, we instantiate this theorem for the q-ary extension, Λq (ρ) = ρh1(q/)(1+ρ)(P) + ρ (R − 1)log2 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.  \nIndex Terms  \nGuesswork, GRAND decoding, random linear codes, Rnyi entropy, guesswork exponent, coset enumeration, transfer theorem, list guesswork, second-order exponent, universality theorem, q-ary guesswork, LDPC ensemble.  \nI. INTRODUCTION  \nThe guesswork of a random variable X introduced by Massey [1] and quantified by Arıkan [2] and Arıkan–Merhav [3] counts how many guesses an optimal strategy requires to identify a realization of X. For an i.i.d. source Xn ∼ P⊗Xn: limn→∞ ~~1~~n logE[G(Xn )ρ] = ρh1ρ (X), an exact equality proved in [2] . The Rnyi entropy hα (X) at order α = 1/(1+ρ) \u003C 1 thus governs the exponential growth rate of the ρ-th guesswork moment. Connections to large deviations and channel coding have been explored in [4], [5] . Guessing Random Additive Noise Decoding (GRAND) [6] decodes by querying noise patterns e′ in decreasing-probability order, testing H (y ⊕ e′)T = 0 at each step. Its query complexity GGRAND ranges over all of {0, 1}n.  \nThe paper is organized as follows. Section II establishes notation. Section III proves the sandwich inequality. Section IV proves the uniform spectrum law. Section V proves the partition-function exponent. Section VI assembles the main theorem. Section VII states the transfer theorem and its consequences. Section VIII derives the list-guesswork exponent. Section IX gives the second-order refinement. Section X proves the universality theorem and the q-ary exponent. Section XI applies the universality theorem to Gallager’s regular LDPC ensemble. Section XII presents numerical validation.  \nII. SYSTEM MODEL AND NOTATION  \nLet n be the blocklength, m = n(1 − R) the number of parity checks, and k = nR the dimension, with R ∈ (0 , 1) . The parity-check matrix H ∈ Fm2×n is drawn uniformly over all full-rank binary matrices of that size. The code is C (H) = {c ∈ Fn2 : HcT = 0} . The noise vector is e ∼ Bernoulli(p)⊗n , p ∈ (0 , ~~1~~2), independent of H, with P (e) = pwH (e)(1 − p)n−wH (e) . Since p \u003C ~~1~~2 , P (e) is strictly decreasing in the Hamming weight wH (e) . The syndrome is σ = HeT ∈ Fm2 . The coset of e is N (H,σ) = {e′ ∈ Fn2 : He′T = σ}, with |N (H,σ)| = 2k when rank(H) = m. The conditional distribution on the coset is Qσ (e′) = P (e′)/Zσ (1), where Zσ (α) = Pe′ ∈N (H,σ) P (e′)α ,α ∈ (0 , 1] . Weight enumerator is Aw (H,σ) =􀀌 􀀈e′ ∈ N (H,σ) : wH (e′) = w 􀀉 􀀌 , ","cbCaifdsjVtRXw3I","https://ap.wps.com/l/cbCaifdsjVtRXw3I","pdf",513410,2,1,13,"English","en",105,"# Introduction\n# System Model and Notation\n## Example: Hamming Code\n## Definitions: Weight Enumerator Growth Rate\n## Definitions: Ln-list Constrained Guesswork\n# Main Results (as outlined)","[{\"question\":\"What quantity does the paper analyze under linear constraints?\",\"answer\":\"It analyzes the ρ-th moment of constrained guesswork, defined as the rank of the true noise vector within its syndrome coset for a random binary linear code.\"},{\"question\":\"How does the constrained guesswork exponent compare to the unconstrained Arıkan–Merhav exponent?\",\"answer\":\"The constrained exponent shifts downward by exactly ρ(1−R) relative to the unconstrained Arıkan–Merhav exponent, with each of the n(1−R) parity checks contributing equally.\"},{\"question\":\"What generalization does the paper prove beyond the binary i.i.d. noise setting?\",\"answer\":\"It proves a universality theorem: for any code ensemble whose weight enumerator concentrates at a rate gE(δ), the guesswork exponent is determined by an expression involving ψα(gE) with a supremum over δ.\"}]",1784188118,33,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"guesswork-under-linear-constraints-exact-exponent-for-coset-decoding","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/guesswork-under-linear-constraints-exact-exponent-for-coset-decoding/83459/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What quantity does the paper analyze under linear constraints?","Question",{"text":75,"@type":76},"It analyzes the ρ-th moment of constrained guesswork, defined as the rank of the true noise vector within its syndrome coset for a random binary linear code.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the constrained guesswork exponent compare to the unconstrained Arıkan–Merhav exponent?",{"text":80,"@type":76},"The constrained exponent shifts downward by exactly ρ(1−R) relative to the unconstrained Arıkan–Merhav exponent, with each of the n(1−R) parity checks contributing equally.",{"name":82,"@type":73,"acceptedAnswer":83},"What generalization does the paper prove beyond the binary i.i.d. noise setting?",{"text":84,"@type":76},"It proves a universality theorem: for any code ensemble whose weight enumerator concentrates at a rate gE(δ), the guesswork exponent is determined by an expression involving ψα(gE) with a supremum over 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