Mathematical Notes

, Volume 104, Issue 5–6, pp 859–865

# On the Recovery of an Integer Vector from Linear Measurements

• S. V. Konyagin
Article

## Abstract

Let 1 ≤ 2lm < d. A vector x ∈ ℤd is said to be l-sparse if it has at most l nonzero coordinates. Let an m × d matrix A be given. The problem of the recovery of an l-sparse vector x ∈ Zd from the vector y = Ax ∈ Rm is considered. In the case m = 2l, we obtain necessary conditions 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 ℤd. For a fixed m, these conditions on d differ only by a logarithmic factor depending on k.

## Keywords

nonsingular matrix lattices successive minima

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