The determinant and rank criteria used for space-time code design apply at high SNR. Code design metrics developed for low SNR assume a channel autocorrelation matrix with equal eigenvalues, which does not hold in many practical scenarios. This paper shows that a space-time code designed to ensure full diversity at high SNR can suffer significant degradation when implemented at low-to-medium SNR because of the channel autocorrelation profile. We examine the effect of the channel autocorrelation matrix on a space-time code's performance and discuss how knowledge of this matrix can be used for code design, particularly from the aspect of space-time trellis code minimum memory requirements. Our discussion applies to both flat-fading and frequency-selective channels that are treated in a unified manner.