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Register a new userSuperimposed Codes and Threshold Group Testing
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Added by
Nikita Polianskii
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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We will discuss superimposed codes and non-adaptive group testing designs arising from the potentialities of compressed genotyping models in molecular biology. The given paper was motivated by the 30th anniversary of Dyachkov-Rykov recurrent upper bound on the rate of superimposed codes published in 1982. We were also inspired by recent results obtained for non-adaptive threshold group testing which develop the theory of superimposed codes
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We consider non-adaptive threshold group testing for identification of up to $d$ defective items in a set of $n$ items, where a test is positive if it contains at least $2 leq u leq d$ defective items, and negative otherwise. The defective items can be identified using $t = O left( left( frac{d}{u} right)^u left( frac{d}{d - u} right)^{d-u} left(u log{frac{d}{u}} + log{frac{1}{epsilon}} right) cdot d^2 log{n} right)$ tests with probability at least $1 - epsilon$ for any $epsilon > 0$ or $t = O left( left( frac{d}{u} right)^u left( frac{d}{d -u} right)^{d - u} d^3 log{n} cdot log{frac{n}{d}} right)$ tests with probability 1. The decoding time is $t times mathrm{poly}(d^2 log{n})$. This result significantly improves the best known results for decoding non-adaptive threshold group testing: $O(nlog{n} + n log{frac{1}{epsilon}})$ for probabilistic decoding, where $epsilon > 0$, and $O(n^u log{n})$ for deterministic decoding.
The goal of threshold group testing is to identify up to $d$ defective items among a population of $n$ items, where $d$ is usually much smaller than $n$. A test is positive if it has at least $u$ defective items and negative otherwise. Our objective is to identify defective items in sublinear time the number of items, e.g., $mathrm{poly}(d, ln{n}),$ by using the number of tests as low as possible. In this paper, we reduce the number of tests to $O left( h times frac{d^2 ln^2{n}}{mathsf{W}^2(d ln{n})} right)$ and the decoding time to $O left( mathrm{dec}_0 times h right),$ where $mathrm{dec}_0 = O left( frac{d^{3.57} ln^{6.26}{n}}{mathsf{W}^{6.26}(d ln{n})} right) + O left( frac{d^6 ln^4{n}}{mathsf{W}^4(d ln{n})} right)$, $h = Oleft( frac{d_0^2 ln{frac{n}{d_0}}}{(1-p)^2} right)$ , $d_0 = max{u, d - u }$, $p in [0, 1),$ and $mathsf{W}(x) = Theta left( ln{x} - ln{ln{x}} right).$ If the number of tests is increased to $Oleft( h times frac{d^2ln^3{n}}{mathsf{W}^2(d ln{n})} right),$ the decoding complexity is reduced to $O left(mathrm{dec}_1 times h right),$ where $mathrm{dec}_1 = max left{ frac{d^2 ln^3{n}}{mathsf{W}^2(d ln{n})}, frac{ud ln^4{n}}{mathsf{W}^3(d ln{n})} right}.$ Moreover, our proposed scheme is capable of handling errors in test outcomes.
In this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the $n times n$ matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain condition on the group ring element $v,$ one can construct non-trivial group LCD codes. Moreover, we also show that by adding more constraints on the group ring element $v,$ one can construct group LCD codes that are reversible. We present many examples of binary group LCD codes of which some are optimal and group reversible LCD codes with different parameters.
In this paper, we propose algorithms that leverage a known community structure to make group testing more efficient. We consider a population organized in disjoint communities: each individual participates in a community, and its infection probability depends on the community (s)he participates in. Use cases include families, students who participate in several classes, and workers who share common spaces. Group testing reduces the number of tests needed to identify the infected individuals by pooling diagnostic samples and testing them together. We show that if we design the testing strategy taking into account the community structure, we can significantly reduce the number of tests needed for adaptive and non-adaptive group testing, and can improve the reliability in cases where tests are noisy.
In this paper, we propose algorithms that leverage a known community structure to make group testing more efficient. We consider a population organized in connected communities: each individual participates in one or more communities, and the infection probability of each individual depends on the communities (s)he participates in. Use cases include students who participate in several classes, and workers who share common spaces. Group testing reduces the number of tests needed to identify the infected individuals by pooling diagnostic samples and testing them together. We show that making testing algorithms aware of the community structure, can significantly reduce the number of tests needed both for adaptive and non-adaptive group testing.
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