This paper presents novel results on scalar feedback quantization
(SFQ) with uniform quantizers. We focus on general SFQ configurations
where reconstruction is via a linear combination of frame
vectors. Using a deterministic approach, we derive two necessary
and sufficient conditions for SFQ to be optimal, i.e., to produce, for
every input, a quantized sequence that is a global minimizer of the
2-norm of the reconstruction error. The first optimality condition is
related to the design of the feedback quantizer, and can always be
achieved. The second condition depends only on the reconstruction
vectors, and is given explicitly in terms of the Gram matrix of the reconstruction
frame. As a by-product, we also show that the the first
condition alone characterizes scalar feedback quantizers that yield
the smallest MSE, when one models quantization noise as uncorrelated,
identically distributed random variables.