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تسجيل مستخدم جديدNon-Adaptive Group Testing Framework based on Concatenation Code
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We consider an efficiently decodable non-adaptive group testing (NAGT) problem that meets theoretical bounds. The problem is to find a few specific items (at most $d$) satisfying certain characteristics in a colossal number of $N$ items as quickly as possible. Those $d$ specific items are called textit{defective items}. The idea of NAGT is to pool a group of items, which is called textit{a test}, then run a test on them. If the test outcome is textit{positive}, there exists at least one defective item in the test, and if it is textit{negative}, there exists no defective items. Formally, a binary $t times N$ measurement matrix $mathcal{M} = (m_{ij})$ is the representation for $t$ tests where row $i$ stands for test $i$ and $m_{ij} = 1$ if and only if item $j$ belongs to test $i$. There are three main objectives in NAGT: minimize the number of tests $t$, construct matrix $mathcal{M}$, and identify defective items as quickly as possible. In this paper, we present a strongly explicit construction of $mathcal{M}$ for when the number of defective items is at most 2, with the number of tests $t simeq 16 log{N} = O(log{N})$. In particular, we need only $K simeq N times 16log{N} = O(Nlog{N})$ bits to construct such matrices, which is optimal. Furthermore, given these $K$ bits, any entry in the matrix can be constructed in time $O left(ln{N}/ ln{ln{N}} right)$. Moreover, $mathcal{M}$ can be decoded with high probability in time $Oleft( frac{ln^2{N}}{ln^2{ln{N}}} right)$. When the number of defective items is greater than 2, we present a scheme that can identify at least $(1-epsilon)d$ defective items with $t simeq 32 C(epsilon) d log{N} = O(d log{N})$ in time $O left( d frac{ln^2{N}}{ln^2{ln{N}}} right)$ for any close-to-zero $epsilon$, where $C(epsilon)$ is a constant that depends only on $epsilon$.
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