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In `Sparse block circulant matrices for compressed sensing', in order to apply Lemma 4, every off-diagonal element of Gram matrix for the sparse block circulant matrix was separated into two component sums to make sure the terms in each sum are independent. In this comment, however, the authors show that separating every element into two sums is not sufficient to guarantee the independency of the terms in each sum. The authors also prove that the entries should be split into three parts instead of two to satisfy the requirements of Lemma 4. Finally, the authors modify the deduction and the result.