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This note is concerned with deterministic constructions of $m \times N$matrices satisfying a restricted isometry property from $\ell_2$ to $\ell_1$ on$s$-sparse vectors. Similarly to the standard ($\ell_2$ to $\ell_2$) restrictedisometry property, such constructions can be found in the regime $m \asymps^2$, at least in theory. With effectiveness of implementation in mind, twosimple constructions are presented in the less pleasing but still relevantregime $m \asymp s^4$. The first one, executing a Las Vegas strategy, isquasideterministic and applies in the real setting. The second one, exploitingGolomb rulers, is explicit and applies to the complex setting. As a steppingstone, an explicit isometric embedding from $\ell_2^n(\mathbb{C})$ to$\ell_4^{cn^2}(\mathbb{C})$ is presented. Finally, the extension of the problemfrom sparse vectors to low-rank matrices is raised as an open question.

Linearly Embedding Sparse Vectors from $\ell_2$ to $\ell_1$ via Deterministic Dimension-Reducing Maps