Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Optimality of the Kautz-Singleton Construction in Probabilistic
  Group Testing   We consider the probabilistic group testing problem where $d$ random
defective items in a large population of $N$ items are identified with high
probability by applying binary tests. It is known that $\Theta(d \log N)$ tests
are necessary and sufficient to recover the defective set with vanishing
probability of error when $d = O(N^{\alpha})$ for some $\alpha \in (0, 1)$.
However, to the best of our knowledge, there is no explicit (deterministic)
construction achieving $\Theta(d \log N)$ tests in general. In this work, we
show that a famous construction introduced by Kautz and Singleton for the
combinatorial group testing problem (which is known to be suboptimal for
combinatorial group testing for moderate values of $d$) achieves the order
optimal $\Theta(d \log N)$ tests in the probabilistic group testing problem
when $d = \Omega(\log^2 N)$. This provides a strongly explicit construction
achieving the order optimal result in the probabilistic group testing setting
for a wide range of values of $d$. To prove the order-optimality of Kautz and
Singleton's construction in the probabilistic setting, we provide a novel
analysis of the probability of a non-defective item being covered by a random
defective set directly, rather than arguing from combinatorial properties of
the underlying code, which has been the main approach in the literature.
Furthermore, we use a recursive technique to convert this construction into one
that can also be efficiently decoded with only a log-log factor increase in the
number of tests.

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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: eess
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph