$C$ is a universal constant independent of both $N$ and $r$ . We show that such matrices are Bounded Orthonormal Systems (BOSs) with known upperbounds on their BOS constants . The new bounds provided herein on themaximum $ell^{\infty$-norms of their $\ell^2-normalized singular vectors allow for several previous Sigma-Delta quantization results to be generalized . Such finite difference matrices alsofundamental to standard $r^{\rm th}$ order Sigma- Delta quantization schemesmore specifically, and as a result the new bounds provide herein on . themaximaximum . of their . normalized singular . norms allow for . several previous . Sigma- delta quantizationresults to be . improved and improved . In

**Author(s) :**Theodore Faust, Mark Iwen, Rayan Saab, Rongrong Wang

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Keywords : sigma - delta - quantization - singular - norms -