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Group Testing with Selectable Thresholds

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper shows that letting each pooled test choose its own positive threshold lets non-adaptive group testing reach the maximum possible rate of one bit per test when the threshold is unbounded.

desk verdict New model and a likely-solid converse, but Theorem 4's achievability rests on a COMP/DD claim that does not hold at the stated test count. read the letter →

arxiv 2607.14448 v1 pith:QIF37T7Y submitted 2026-07-16 cs.IT math.ITmath.PR

Trung-Khang Tran , Daniel McMorrow , Jonathan Scarlett This is my paper
classification cs.ITmath.ITmath.PR MSC 94A15
keywords grouptestingselectablethresholdsthresholdnon-adaptivealgorithmscountingboundachievableratesconverseboundspooled
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces a variant of group testing in which each pooled test comes with a threshold chosen by the designer: the test is positive if and only if the number of defective items in it is at least that threshold. Its central finding is that this flexibility is powerful: with no cap on the threshold, non-adaptive strategies can recover the defective set with a number of tests equal to k log2(n/k)(1+o(1)), reaching rate 1 and matching the counting bound that no binary-outcome strategy can beat. With a fixed maximum threshold, the paper gives a two-stage polynomial-time algorithm with an explicit test-count formula, and a matching converse under regularity conditions in the dense limit k = Theta(n^theta), theta -> 1. A sympathetic reader would care because it sharpens what kinds of 1-bit test outcomes are useful, shows selectable thresholds can beat standard group testing in the dense regime, and clarifies that the largest threshold dominates information-theoretically.

What carries the argument

The two-batch algorithm is the central construction: batch one uses standard threshold-1 group testing (COMP/DD, or a SUBSET subroutine in the unbounded case) to find a set S_DD of k(1-o(1)) defectives plus a superset containing all defectives; batch two runs near-constant-column-weight tests at threshold gamma_max, and an item is 'resolved' by a test containing exactly gamma_max-1 items from S_DD and no other uncertain item. The probability of such a test is asymptotically Poisson, nu^{gamma_max-1} e^{-nu}/(gamma_max-1)!, which produces the explicit denominator in the rate formula. The converse rests on the masking mechanism: an item is masked when no test is informative for it (contains it

What would settle it

Run the fixed-gamma_max two-batch algorithm for gamma_max=2 with n=10^6 and k=n^0.95, sweep T around the predicted value, and check whether empirical error probability drops to zero at the predicted (1/log 2) k log2(n/k) + k log k / 0.636 tests; a mismatch would falsify Theorem 4, while agreement would support the claim that selectable thresholds strictly beat standard group testing in the dense regime.

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Extended reading notes

Core claim

The paper's central claim is that making the threshold a selectable parameter turns threshold group testing into a problem whose information-theoretic rate can reach 1 when gamma_max is unbounded, and whose fixed-threshold rate is characterized by matching achievability and converse bounds in the dense limit. The constructive idea is a two-batch scheme: first use ordinary threshold-1 group testing to identify a near-complete set of defectives, then use tests with threshold gamma_max to resolve the remaining uncertain items; the fixed-threshold analysis shows the second batch needs k log k / [nu log(1/(1 - nu^{gamma_max-1} e^{-nu}/(gamma_max-1)!))] tests. On the converse side, the paper prove

Load-bearing premise

The rate-1 and large-gamma_max achievability claims rest on a cited SUBSET subroutine that the paper flags as only proved for a computationally inefficient method, and the matching converse additionally assumes near-constant test sizes/degrees with bounded overlaps; if either premise gives way, the corresponding headline results remain conditional.

Editorial extensions

If this is right

  • With unbounded selectable thresholds, T = k log2(n/k)(1+o(1)) tests suffice for non-adaptive recovery, matching the counting bound and making the rate exactly 1.
  • For fixed gamma_max, the two-batch algorithm runs in polynomial time and uses (1/log 2) k log2(n/k) + min_nu k log k / [nu log(1/(1 - nu^{gamma_max-1} e^{-nu}/(gamma_max-1)!))] (1+o(1)) tests.
  • As gamma_max grows, the achievable rate approaches 1 at least as fast as 1 - O(1/sqrt(gamma_max)), and the converse shows the k log k coefficient cannot decay faster than 1/sqrt(gamma_max).
  • In the dense limit theta -> 1, achievability and converse for fixed gamma_max have matching k log k coefficient, so the number of tests is determined to within o(k log k).
  • Information-theoretically, selectable thresholds do not beat just using the largest threshold; their value is algorithmic simplicity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The two-batch template suggests a general recipe: any standard group-testing method that returns a near-complete defective set can be upgraded with high-threshold resolution tests, and nothing in the analysis appears to require exactly two batches.
  • Because the converse assumes near-constant weights/degrees and bounded overlaps, the picture for non-regular designs such as Bernoulli-type test matrices is open; a numerical study in the dense limit would show whether the matched k log k coefficient survives without those regularity conditions.
  • The proven rates use only thresholds 1 and gamma_max; a natural testable extension is whether intermediate thresholds close the remaining gap for fixed theta < 1, although the paper's Lemma 1 suggests no information-theoretic gain from them.
  • For applications with noisy threshold devices, the idealized exact-threshold model would need calibration, and the second batch's reliance on exact counts of gamma_max-1 known defectives suggests a noise sensitivity that could be probed by simulation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces selectable-threshold group testing (GT-ST): each binary pooled test can be assigned its own threshold γ(t) ≤ γmax, with outcome 1{ψ_t ≥ γ(t)}. The authors prove the counting bound; show rate 1 is achievable when γmax is unbounded (Theorem 2) and rate 1−O(γmax^{-1/2}) for large γmax (Theorem 3), both by a two-batch strategy built on the SUBSET result of [22]; give a finite-γmax achievability result with polynomial-time COMP/DD decoding (Theorem 4); and prove masking-based converses (Theorems 5 and 6) under near-constant weight/degree and bounded-overlap assumptions, with an optimization lemma showing that the converse is dominated by the largest threshold. The proofs are detailed and the exposition is careful.

Significance. If its results held, the paper would be a valuable contribution: it formulates a natural generalization of threshold group testing, gives simple rate-1 constructions in the unbounded/large-threshold regime (modulo the acknowledged external SUBSET subroutine), and provides a converse that is more general than concurrent work. The masking-based proof is sophisticated and mostly internally consistent, and the authors are explicit about assumptions and external dependencies. However, the central finite-γmax achievability theorem is unsupported as written because the batch-1 COMP/DD guarantee is quantitatively false at the stated test count; this directly undermines Eq. (11) and the claimed comparison with standard GT.

major comments (2)
  1. [App. B-C, Theorem 4, Eq. (11)] The batch-1 claim that COMP produces S_COMP of size k(1+o(1)) is not a valid consequence of [22, App. D] and is contradicted by the authors' own parameters. With T1=(1/log2)k log2(n/k) and L1=T1 log2/k=log2(n/k), the effective inclusion probability is p=L1/T1=log2/k. For a fixed non-defective, a test is negative with probability (1-p)^k→1/2, so the expected number of negative tests containing the item is T1·p/2=(1/2)log2(n/k). Hence P(item remains in S_COMP)=exp(−(1/2)log2(n/k))=(n/k)^{−1/(2 ln2)}≈(n/k)^{−0.721}, and E|S_COMP\S|≈n(n/k)^{−0.721}=n^{0.279+0.721θ}, which is ω(k) for every θ<1. Thus S_COMP cannot have size k(1+o(1)), and the later union bound over |S_COMP\S_DD|=o(k) is invalid.
  2. [Section II-C and Fig. 1] The finite-γmax claims of strict improvement over standard GT are not supported independently of the faulty batch-1 step. At the inclusion probability used in the proof, the number of batch-1 tests must be increased by a factor greater than 2ln2≈1.386 even to make the COMP false positives o(k); the DD step is also contaminated by the large false-PD set. The authors should either supply a correct batch-1 subroutine that provably meets the stated T1, or restate Theorem 4 and Figure 1 with a corrected leading term and re-evaluate the comparison.
minor comments (4)
  1. [Lemmas 2 and 9] The display γ^{-1/3} appears where the proof clearly means γ−1/3 (γ minus one third). Please fix the notation; as typeset it is ambiguous and initially reads as a tiny exponent.
  2. [App. B-C, [22, App. D]] Since [22, App. D] supplies the crucial COMP/DD guarantee for Theorem 4, the paper should state the exact lemma used, including the hypotheses and the precise sense in which S_DD and S_COMP have the claimed sizes, rather than only citing it.
  3. [Fig. 1] The caption should specify precisely which curve from [20] is plotted and in which metric. The current description makes the claimed comparison in the dense limit difficult to verify.
  4. [Theorem 3 proof] The step that discards tests whose repeated outcomes are all identical should clarify that this is valid only on the high-probability event that ψ_t lies in the interval (γ_-, γ_+); a sentence about boundary effects near γ_± would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivations reduce to prior standard-GT theorems, not to their own conclusions.

full rationale

I found no step in which a claimed prediction is equivalent by construction to an input or fitted value. The strongest claims (Theorems 2 and 3) use the SUBSET subroutine from [22] ('This can be done with a number of tests meeting the counting bound [22, Thm. 2]'), and Theorem 4 uses the COMP/DD approximation guarantee from [22, App. D]. These are self-citations (McMorrow and Scarlett are co-authors), and Section I-C is transparent that the SUBSET route is 'albeit only currently proved for a computationally inefficient method'; but the cited results are about standard group testing with one-sided error, not about the selectable-threshold model of this paper. They are used as black-box subroutines, not as a restatement of the target result, so they are independent support rather than circularity. The converse (Theorems 5-6) is derived in the paper from masking and informative-test arguments under Assumptions 1-3; those assumptions are regularity conditions, not conclusions, and are checked for the NCC design in Appendix D. The optimization over nu in Theorem 4 and Delta in Theorem 6 is minimization, not data fitting, so no 'fitted input called prediction' pattern occurs. The dense-limit match between Theorem 4 and Theorem 6 arises from independently derived formulas rather than one being inserted into the other. The skeptic's numerical false-positive calculation against Theorem 4 is a potential correctness gap in a cited COMP/DD claim, but it does not exhibit a circular reduction; it alleges the cited result is wrong at the stated test count, which is a validity concern, not a self-referential derivation. Accordingly no circular step is identified.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. The main external inputs are the prior SUBSET/one-sided-recovery results of [22] and the regularity assumptions on test designs. All numerical parameters (nu, Delta, alpha_r, inclusion probabilities) are optimization or design parameters rather than quantities fitted to empirical data.

free parameters (4)
  • nu (batch-2 inclusion parameter) = optimized over nu>0; for gamma_max=2 the denominator max is numerically about 0.636
    Controls the NCC design in the second batch of Theorem 4; the theorem takes a minimum over nu to optimize the rate.
  • Delta (row-weight constant in converse) = arg max over Delta>0 of h(Delta,gamma); for large gamma, Delta* is in [gamma-1/3, gamma]
    Represents the asymptotic test weight Delta n/k in Assumption 1/3; the converse minimizes over Delta.
  • alpha_r (fraction of tests per threshold) = optimized over the simplex; Lemma 1 shows the maximum is attained by putting all mass on the highest threshold
    Used in the selectable-threshold converse to split tests among allowed thresholds; optimized to minimize the number of tests.
  • Bernoulli inclusion probabilities (log^4 k/k and gamma_max/(2k)) = chosen per theorem: log^4 k/k for unbounded threshold, gamma_max/(2k) for large gamma_max
    Design parameters in Theorems 2 and 3 chosen to make the number of defectives per test concentrate in a desired window.
assumptions (6)
  • domain assumption Uniform combinatorial prior on the defective set: S is uniform over all k-subsets of [n].
    Stated in Section I-A and used throughout for both achievability and converse.
  • domain assumption Sublinear sparsity k = Theta(n^theta) with theta in (0,1).
    Stated in Section I-A; all asymptotic results are in this regime.
  • domain assumption Noiseless threshold measurements: y_t = 1{psi_t >= gamma^(t)} with exactly specified thresholds.
    The idealized model of Section I-A; noise and threshold implementation error are explicitly out of scope.
  • standard math SUBSET achievability at the counting bound from [22].
    Theorems 2 and 3 assume the SUBSET subroutine finds k - k/log^4 k defectives using (k log_2(n/k))(1+o(1)) tests; this is cited to [22] and not reproved.
  • ad hoc to paper Assumptions 1-3: near-constant item degrees and test weights, and bounded pairwise test intersections.
    The converse theorems require these regularity conditions on deterministic designs; the authors verify them for the NCC design and state that the general case is open.
  • standard math Standard probabilistic tools: Chernoff, McDiarmid, FKG, Poisson approximation, Stirling bounds.
    Used throughout the appendices for concentration and decoupling arguments.

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Pith. "Pith review of Group Testing with Selectable Thresholds." pith.science (2026). https://pith.science/paper/QIF37T7Y

@misc{pith2026260714448,
  author       = {Pith},
  title        = {Pith review of: Group Testing with Selectable Thresholds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QIF37T7Y}},
  note         = {Machine review of arXiv:2607.14448}
}
abstract

We consider the problem of group testing, in which one seeks to identify a subset of defective items of size $k$ from a larger set of $n$ items based on pooled tests. We introduce a selectable threshold model, in which each test has an associated threshold that can be chosen, such that the test outcome is 1 if and only if the number of defectives in the test is no smaller than that threshold. In settings with a large or unbounded maximum threshold, we establish conditions under which high-probability recovery can be attained with a rate (i.e., the asymptotic ratio of $\log_2{n \choose k}$ to the number of tests) approaching its maximum possible value of 1. Moreover, in the case of a fixed maximum threshold, we establish an achievable number of tests using simple and computationally efficient decoding methods, and a converse that holds under suitable regularity conditions on the test design, with the two coinciding in the dense limit (i.e., $\theta$ approaching one in the scaling $k = \Theta(n^{\theta})$).

Figures

Figures reproduced from arXiv: 2607.14448 by the authors.

Figure 1
Figure 1. Comparison of rates with γmax = 10. The achievability curve is from Theorem 4, the converse from Theorem 6 (combined with Lemma 1), and the standard GT curve from [20]. • Beyond the simple counting bound, we establish a refined converse for the finite γmax under suitable regularity conditions on the test matrix. Among other things, this establishes that our achievable number of tests is tight under these regularity … view at source ↗

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Works this paper leans on

43 extracted references · 3 linked inside Pith

  1. [22]

    Optimal non-adaptive group testing with one-sided error guarantees,

    D. McMorrow and J. Scarlett, “Optimal non-adaptive group testing with one-sided error guarantees,”IEEE Transactions on Information Theory, vol. 72, no. 5, pp. 3497–3517, 2026

    2026

  2. [30]

    An information-theoretic analysis of threshold group testing,

    R. van der Hofstad, N. Müller, and C. Riddlesden, “An information-theoretic analysis of threshold group testing,”arXiv preprint arXiv:2606.11353, 2026

    arXiv 2026

  3. [1]

    Pooled testing and its applications in the COVID-19 pandemic,

    M. Aldridge and D. Ellis, “Pooled testing and its applications in the COVID-19 pandemic,” inPandemics: Insurance and Social Protection, pp. 217–249, Springer, 2021

    2021

  4. [2]

    Pooling dna in the identification of parents,

    R. N. Curnow and A. P. Morris, “Pooling dna in the identification of parents,”Heredity, vol. 80, no. 1, pp. 101–109, 1998

    1998

  5. [3]

    A pooling strategy for heterozygote screening of the ∆F508 cystic fibrosis mutation,

    C. Gille, K. Grade, and C. Coutelle, “A pooling strategy for heterozygote screening of the ∆F508 cystic fibrosis mutation,”Human genetics, vol. 86, no. 3, pp. 289–291, 1991

    1991

  6. [4]

    What’s hot and what’s not: Tracking most frequent items dynamically,

    G. Cormode and S. Muthukrishnan, “What’s hot and what’s not: Tracking most frequent items dynamically,”ACM Transactions on Database Systems (TODS), vol. 30, no. 1, pp. 249–278, 2005

    2005

  7. [5]

    An adaptive technique for local distribution,

    J. Hayes, “An adaptive technique for local distribution,”IEEE Transactions on Communications, vol. 26, no. 8, pp. 1178–1186, 1978

    1978

  8. [6]

    Born again group testing: Multiaccess communications,

    J. Wolf, “Born again group testing: Multiaccess communications,”IEEE Transactions on Information Theory, vol. 31, no. 2, pp. 185–191, 1985

    1985

Show all 43 references
  1. [7]

    Optimal nested test plan for combinatorial quantitative group testing,

    C. Wang, Q. Zhao, and C.-N. Chuah, “Optimal nested test plan for combinatorial quantitative group testing,”IEEE Transactions on Signal Processing, vol. 66, no. 4, pp. 992–1006, 2018

    2018

  2. [8]

    Quantitative group testing and the rank of random matrices,

    U. Feige and A. Lellouche, “Quantitative group testing and the rank of random matrices,”arXiv preprint arXiv:2006.09074, 2020

    2006 arXiv

  3. [9]

    Semiquantitative group testing,

    A. Emad and O. Milenkovic, “Semiquantitative group testing,”IEEE Transactions on Information Theory, vol. 60, no. 8, pp. 4614–4636, 2014

    2014

  4. [10]

    Semiquantitative group testing in at most two rounds,

    M. Cheraghchi, R. Gabrys, and O. Milenkovic, “Semiquantitative group testing in at most two rounds,” inIEEE International Symposium on Information Theory (ISIT), pp. 1973–1978, 2021

    1973

  5. [11]

    Threshold group testing,

    P. Damaschke, “Threshold group testing,” inGeneral Theory of Information Transfer and Combinatorics, pp. 707–718, Springer, 2006

    2006

  6. [12]

    Nonadaptive algorithms for threshold group testing,

    H.-B. Chen and H.-L. Fu, “Nonadaptive algorithms for threshold group testing,”Discrete Applied Mathematics, vol. 157, no. 7, pp. 1581–1585, 2009

    2009

  7. [13]

    Stochastic threshold group testing,

    C. L. Chan, S. Cai, M. Bakshi, S. Jaggi, and V . Saligrama, “Stochastic threshold group testing,” inInformation Theory Workshop (ITW), pp. 1–5, 2013

    2013

  8. [14]

    The capacity of adaptive group testing,

    L. Baldassini, O. Johnson, and M. Aldridge, “The capacity of adaptive group testing,” inIEEE International Symposium on Information Theory (ISIT), pp. 2676–2680, 2013

    2013

  9. [15]

    Group testing: An information theory perspective,

    M. Aldridge, O. Johnson, and J. Scarlett, “Group testing: An information theory perspective,”Foundations and Trends® in Communications and Information Theory, vol. 23, no. 1-2, pp. 1–221, 2026

    2026

  10. [16]

    Group testing algorithms: Bounds and simulations,

    M. Aldridge, L. Baldassini, and O. Johnson, “Group testing algorithms: Bounds and simulations,”IEEE Transactions on Information Theory, vol. 60, no. 6, pp. 3671–3687, 2014

    2014

  11. [17]

    Phase transitions in group testing,

    J. Scarlett and V . Cevher, “Phase transitions in group testing,” inACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 40–53, 2016

    2016

  12. [18]

    Performance of group testing algorithms with near-constant tests per item,

    O. Johnson, M. Aldridge, and J. Scarlett, “Performance of group testing algorithms with near-constant tests per item,”IEEE Transactions on Information Theory, vol. 65, no. 2, pp. 707–723, 2018

    2018

  13. [19]

    Information-theoretic and algorithmic thresholds for group testing,

    A. Coja-Oghlan, O. Gebhard, M. Hahn-Klimroth, and P. Loick, “Information-theoretic and algorithmic thresholds for group testing,”IEEE Transactions on Information Theory, vol. 66, no. 12, pp. 7911–7928, 2020

    2020

  14. [20]

    Optimal group testing,

    A. Coja-Oghlan, O. Gebhard, M. Hahn-Klimroth, and P. Loick, “Optimal group testing,” inConference on Learning Theory, vol. 125 of Proceedings of Machine Learning Research, pp. 1374–1388, PMLR, 2020. July 17, 2026 DRAFT 48

    2020

  15. [21]

    How little does non-exact recovery help in group testing?,

    J. Scarlett and V . Cevher, “How little does non-exact recovery help in group testing?,” inIEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2017

    2017

  16. [23]

    Bounds for threshold and majority group testing,

    R. Ahlswede, C. Deppe, and V . Lebedev, “Bounds for threshold and majority group testing,” inIEEE International Symposium on Information Theory, pp. 69–73, 2011

    2011

  17. [24]

    Improved constructions for non-adaptive threshold group testing,

    M. Cheraghchi, “Improved constructions for non-adaptive threshold group testing,”Algorithmica, vol. 67, no. 3, pp. 384–417, 2013

    2013

  18. [25]

    Improved non-adaptive algorithms for threshold group testing with a gap,

    T. V . Bui, M. Cheraghchi, and I. Echizen, “Improved non-adaptive algorithms for threshold group testing with a gap,”IEEE Transactions on Information Theory, vol. 67, no. 11, pp. 7180–7196, 2021

    2021

  19. [26]

    Subquadratic non-adaptive threshold group testing,

    G. De Marco, T. Jurdzi ´nski, M. Ró ˙za´nski, and G. Stachowiak, “Subquadratic non-adaptive threshold group testing,” inInternational Symposium on Fundamentals of Computation Theory, pp. 177–189, Springer, 2017

    2017

  20. [27]

    Efficiently decodable non-adaptive threshold group testing,

    T. V . Bui, M. Kuribayashi, M. Cheraghchi, and I. Echizen, “Efficiently decodable non-adaptive threshold group testing,”IEEE Transactions on Information Theory, vol. 65, no. 9, pp. 5519–5528, 2019

    2019

  21. [28]

    Efficient designs for threshold group testing without gap,

    T. V . Bui, Y . M. Chee, and V . K. Vu, “Efficient designs for threshold group testing without gap,” inIEEE International Symposium on Information Theory (ISIT), pp. 3005–3010, 2024

    2024

  22. [29]

    Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities,

    T. Laarhoven, “Asymptotics of fingerprinting and group testing: Tight bounds from channel capacities,”IEEE Transactions on Information Forensics and Security, vol. 10, no. 9, pp. 1967–1980, 2015

    1967

  23. [31]

    Algorithms for threshold group testing,

    A. Coja-Oghlan, R. van der Hofstad, L. Krieg, N. Müller, C. Riddlesden, and O. Scheftelowitsch, “Algorithms for threshold group testing,” arXiv preprint arXiv:2606.27127, 2026

    2026 arXiv

  24. [32]

    Optimal non-adaptive probabilistic group testing in general sparsity regimes,

    W. H. Bay, J. Scarlett, and E. Price, “Optimal non-adaptive probabilistic group testing in general sparsity regimes,”Information and Inference: A Journal of the IMA, vol. 11, no. 3, pp. 1037–1053, 2022

    2022

  25. [33]

    Individual testing is optimal for nonadaptive group testing in the linear regime,

    M. Aldridge, “Individual testing is optimal for nonadaptive group testing in the linear regime,”IEEE Transactions on Information Theory, vol. 65, no. 4, pp. 2058–2061, 2018

  26. [34]

    Boolean compressed sensing and noisy group testing,

    G. K. Atia and V . Saligrama, “Boolean compressed sensing and noisy group testing,”IEEE Transactions on Information Theory, vol. 58, no. 3, pp. 1880–1901, 2012

    1901

  27. [35]

    Mitzenmacher and E

    M. Mitzenmacher and E. Upfal,Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis. Cambridge University Press, 2017

    2017

  28. [36]

    On the method of bounded differences,

    C. McDiarmid, “On the method of bounded differences,” inBritish Combinatorial Conference, p. 148–188, 1989

    1989

  29. [37]

    Grimmett,Percolation

    G. Grimmett,Percolation. Springer, 2 ed., 1999

    1999

  30. [38]

    R. B. Ash,Information theory. Courier Corporation, 1990

    1990

  31. [39]

    Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms,

    C. L. Chan, P. H. Che, S. Jaggi, and V . Saligrama, “Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms,” inAllerton Conference on Communication, Control, and Computing, pp. 1832–1839, IEEE, 2011

    2011

  32. [40]

    A comparison of signalling alphabets,

    E. N. Gilbert, “A comparison of signalling alphabets,”The Bell System Technical Journal, vol. 31, no. 3, pp. 504–522, 1952

    1952

  33. [41]

    Bounds for the Gamma function,

    N. Batir, “Bounds for the Gamma function,”Results in Mathematics, vol. 72, no. 1, pp. 865–874, 2017

    2017

  34. [42]

    D. P. Bertsekas,Nonlinear Programming. Belmont, Massachusetts: Athena Scientific, 2 ed., 1999

    1999

  35. [43]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, eds.,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. No. 55 in National Bureau of Standards, Applied Mathematics Series, 1964. (Tenth printing, Dec. 1972). July 17, 2026 DRAFT

    1964

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