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This paper studies a quantized implementation of a Gramian-based distributed observer for discrete-time systems observed by a network of sensor nodes. The starting point is a distributed observer based on local information matrices obtained from a distributed constructibility Gramian recursion. In order to account for limited communication, the information matrices and information vectors exchanged among neighboring nodes are quantized by means of a uniform quantizer with saturation. The resulting algorithm preserves the original distributed structure and requires only one communication round per sampling instant. A simple perturbation argument shows that the estimation error induced by quantization depends on the perturbation of the local information matrix and information vector, and therefore on the quantization resolution. Numerical results illustrate a sharp practical threshold on the number of communication bits: in the considered example the estimator remains stable for 25 bits, while 23 bits already lead to divergence.