@article{Seong2026, 
author = {Jin-Taek Seong},
title = {Finite-field pooling for community state inference: sample complexity bounds at the saturation point},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {18502-18524},
keywords = {community state inference, finite fields, compressive sensing, information-theoretic bounds, MAP decoding},
url = {https://www.sciopen.com/article/10.3934/math.2026752},
doi = {10.3934/math.2026752},
abstract = {We studied the sample complexity of community state inference, in which a    K-sparse latent state vector        x    ∈            F        q    N   over a known community partition is to be recovered from pooled observations        y    =      A        x   over the finite field              F        q  . The pooling matrix        A   has a constant row weight of    d, with modeling pools formed as linear combinations of exactly    d members. We derived necessary and sufficient conditions on the number of pooled observations    M. Let        α    t    (  d  ) denote the probability that a row of        A   misses a fixed set of size    t. The lower bound, obtained from Fano's inequality, is governed by        α    K    (  d  ); the upper bound, obtained under maximum a posteriori (MAP) decoding, is governed by        α          2      K        (  d  ), thus reflecting the worst-case overlap between two candidate supports of total size    2  K. Under the sparse-regime approximation        α          2      K        (  d  )  ≈      e          −      2      K      d              /            N      , we identified the asymptotic saturation point        d          ∗        =  (  N      /    (  2  K  )  )  ln  ⁡  q as the solution of        α          2      K        (  d  )  =  1      /    q, at which the two bounds match to order    Θ  (  K      log    q    ⁡  (  N      /    K  )  ) with a multiplicative gap bounded by a constant    C  (  q  ) that satisfies    C  (  q  )  →  2 as    q  →  ∞. The analysis was restricted to the sparse signal regime    K  →  ∞,    K      /    N  →  0,    log  ⁡  q  =  o  (  K  ), and assumes noiseless observations; no practical decoder is proposed.}
}