In this passage, Hoppe practically takes the Lange-Lerner position on knowledge and prices:
"the knowledge conveyed by prices actually can be centralized. But if price information is public information and thus can be centralized, then, according to Hayek's thesis that socialism's problem stems from the inefficiency of trying to centralize genuinely uncentralizable private knowledge, it would follow that the absence of prices, and hence of private property, has nothing to do with the plight of socialism."
This presents Hayek's arguments in essentially the same terms as Oliver Williamson's theory of the firm. To a certain extent I would agree with that characterization of Hayek, although I wouldn't consider it to be as unsatisfactory as Hoppe apparently does. The "knowledge problem" version of the "calculation problem" does turn on the practical inefficiency of socialism, rather than the logical impossibility of socialism (I can agree with Hoppe on that). But the Hayekian argument loses none of its force for being an accusation of practical inefficiency, rather than logical impossibility (here I disagree with Hoppe).
Hoppe prefers what he contends is the Misesian version of the argument (I have to take his word on that - I've never read Mises on the calculation problem). Unfortunately, this essay is almost entirely taken up with Hoppe's critique of Hayek, so little is offered in elaborating this property-based position. This is all we really have in the essay in terms of an exposition of his own position:
"Mises's well-known calculation argument states this: If there is no private property in land and other production factors, then there can also be no market prices for them. Hence, economic calculation, i.e., the comparison, in light of current prices, of anticipated revenue, and expected cost expressed in terms of a common medium of exchange-money-(thus permitting cardinal accounting operations), is literally impossible. Therefore, socialism's fatal error is the absence of private property in land and production factors, and, by implication, the absence of economic calculation."