Geodesic
On the Recovery of an Integer Vector from Linear Measurements
Matematičeskie zametki, Tome 104 (2018) no. 6, pp. 863-871 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $1\le 2l\le m. A vector $x\in\mathbb Z^d$ is said to be $l$-sparse if it has at most $l$ nonzero coordinates. Let an $m\times d$ matrix $A$ be given. The problem of the recovery of an $l$-sparse vector $x\in\mathbb Z^d$ from the vector $y=A x\in\mathbb R^m$ is considered. In the case $m=2l$, we obtain necessary and sufficient conditions on the numbers $m$, $d$, and $k$ ensuring the existence of an integer matrix $A$ all of whose elements do not exceed $k$ in absolute value which makes it possible to reconstruct $l$-sparse vectors in $\mathbb Z^d$. For a fixed $m$, these conditions on $d$ differ only by a logarithmic factor depending on $k$.
Keywords: nonsingular matrix, lattices, successive minima. 
@article{MZM_2018_104_6_a5,
     author = {S. V. Konyagin},
     title = {On the {Recovery} of an {Integer} {Vector} from {Linear} {Measurements}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {863--871},
     year = {2018},
     volume = {104},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_104_6_a5/}
}
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S. V. Konyagin. On the Recovery of an Integer Vector from Linear Measurements. Matematičeskie zametki, Tome 104 (2018) no. 6, pp. 863-871. http://geodesic.mathdoc.fr/item/MZM_2018_104_6_a5/

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