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We present new constraints on the 21cm Epoch of Reionization (EoR) powerspectrum derived from 3 months of observing with a 32-antenna,dual-polarization deployment of the Donald C. Backer Precision Array forProbing the Epoch of Reionization (PAPER) in South Africa. In this paper, wedemonstrate the efficacy of the delay-spectrum approach to avoidingforegrounds, achieving over 8 orders of magnitude of foreground suppression (in$\textrm{mK}^2$). Combining this approach with a procedure for removingoff-diagonal covariances arising from instrumental systematics, we achieve abest 2-sigma upper limit of $(41\,\textrm{mK})^2$ for $k=0.27h\textrm{Mpc}^{-1}$ at $z=7.7$. This limit falls within an order of magnitudeof the brighter predictions of the expected 21cm EoR signal level. Using theupper limits set by these measurements, we generate new constraints on thebrightness temperature of 21cm emission in neutral regions for variousreionization models. We show that for several ionization scenarios, ourmeasurements are inconsistent with cold reionization. That is, heating of theneutral intergalactic medium (IGM) is necessary to remain consistent with theconstraints we report. Hence, we have suggestive evidence that by $z=7.7$, theHI has been warmed from its cold primordial state, probably by X-rays fromhigh-mass X-ray binaries or mini-quasars. The strength of this evidence dependson the ionization state of the IGM, which we are not yet able to constrain.This result is consistent with standard predictions for how reionization mighthave proceeded.

New Limits on 21cm EoR From PAPER-32 Consistent with an X-Ray Heated IGM at z=7.7