Properties of the MMSE of “bad” codes

We examine non-optimal codes (alternatively referred to as “bad” codes), over the additive Gaussian noise channel. These codes are required to attain a minimum rate at a specific signal-to-noise ratio (snr). For these codes, we provide a tight lower bound on the minimum mean square error (MMSE), valid for any snr, attainable by superposition codebooks, optimal for a specific degraded Gaussian broadcast channel (BC). Moreover, the MMSE function of codes, attaining a minimum required rate at some snr, and the lower bound on the MMSE at some other snr, is completely defined for all snr, and is the one obtained by the corresponding superposition codebooks.

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