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We propose a technique to derive upper bounds on Gallager's cost-constrained random coding exponent function. Applying this technique to the noncoherent peak-power or average-power limited discrete time memoryless Ricean fading channel, we obtain the high signal-to-noise ratio (SNR) expansion of this channel's cutoff rate. At high SNR, the gap between channel capacity and the cutoff rate approaches a finite limit. This limit is approximately 0.26 nats per channel-use for zero specular component (Rayleigh) fading and approaches 0.39 nats per channel-use for very large values of the specular component. We also compute the asymptotic cutoff rate of a Rayleigh-fading channel when the receiver has access to some partial side information concerning the fading. It is demonstrated that the cutoff rate does not utilize the side information as efficiently as capacity, and that the high SNR gap between the two increases to infinity as the imperfect side information becomes more and more precise.