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Linear coding schemes for noiseless feedback channels have been well studied. These schemes have been shown to display favorable properties for a broad class of Gaussian noise processes. However, the design of linear codes for noisy feedback channels is much less developed. In this work, an analytical expression for the optimal linear code for the additive white Gaussian noise feedback channel is derived. Also, bounds are obtained on the post-processed signal-to-noise ratio for communication systems with similar noise statistics in the forward and the feedback channel. For the general case of arbitrary forward and feedback noisy channels, an iteratively optimized linear coding scheme is proposed.