The "can + pefect infinitive" construction doesn't seem to be impossible, just very rare (probably for semantic reasons, as Janus Bahs Jacquet says). I did a wildcard search on the Google Ngram Viewer to get an idea of the most common sequences of "can have" + verb; it gave, from most to least frequent, the following constructions with a past participle: can have been, can have had, can have done, can have given, can have taken, can have failed, can have happened, can have made, can have become.
Alternative structures that look the same
As Tushar Raj points out, there are other structures that look the same, but have different grammar.
It would be unremarkable if the structure is "can have" + NP, where the noun phrase consists of an "-ed/-en" word before a noun. (Then again, it is my understanding that some people say that an "-ed/-en" word can't even be a participle when it occurs as a pre-nominal modifier, and has to be an adjective in this position.)
It also would be unremarkable if the structure is "can" + "have past-participle" using the "have something done" causative struture. I found the following example:
As for printing, what you can't do in your office, you can have done by professional printers.
Start Your Own Tutoring and Test Prep Business, by Entrepreneur Press and Rich Mintzer
perfect "can have been"
Looking through the Google Books results for examples, a lot of the first examples of perfect "can have been"s were from grammar books, but I did find the following examples of usage "in the wild" so to speak:
With the exception of the few that have owed their rise to trade, think whether among those families which we behold seated on the summit of that eminence which is composed of power, opulence, and factitious dignity, there can have been any who have been raised by anything better depredation: licensed and irresistible depredation; depredation by that swarm of harpies...
– "On Spanish and Portuguese Affairs," in Works, Volume 8, by Jeremy Bentham, published under the superintendence of his executor John Bowring
Thus, I believe you can see that to be “authentic,” a tape recording must include a complete set of events, and nothing can have been added, deleted or changed at any time during the recording or subsequently.
– The Acoustics of Crime: The New Science of Forensic Phonetics, by Harry Hollien
perfect "can have done"
The geese got together in a tight group, stuck their heads together and asked: “ Who can have done this? Who can have done this?”
– The Wonderful Adventures of Nils, by Selma Lagerlöf, translated by Velma Swanston Howard
I believe "could have done" is also possible in this kind of context, maybe even more frequent.
So when Alison Carter went missing in December 1963, it meant more to me and my classmates than it can have done to most other people.
– A Place of Execution, by Val McDermid
I won't go through all the verbs the Ngram Viewer lists ... hopefully these examples are enough to get an idea of how this structure is used. It seems to me that it is used mainly in questions or embedded questions (like the example with "whether"), or in contrast with another verb phrase in a more usual tense.