"The ﬁrst deterministic construction of compressed sensing measurement matrices with an order-optimal number of measurements."

The paper is LDPC Codes for Compressed Sensing by Alexandros Dimakis, Roxana Smarandache, and Pascal Vontobel, The construction is in Corollary 18:

Let dv, dc ∈ Z>0. Consider a measurement matrix HCS ∈ {0, 1} m×n whose Tanner graph is a (dv, dc)-regular bipartite graph with Ω(log n) girth. This measurement matrix succeeds in recovering a randomly supported k = αn sparse vector with probability 1 − o(1) if α is below some threshold function α'(dv, dc, m/n).

Also of related interest is Sparse Binary Matrices of LDPC codes for Compressed Sensing by Weizhi Lu ,Kidiyo Kpalma , Joseph Ronsin. but the authors do not seem to be aware of the LDPC codes for CS paper.mentioned above.

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