The encoding circuit from that paper is weirdly complicated, with things like CSWAPs when only Clifford operations are required (Update: talking with Dave Bacon, I learned this paper predates our conventions on how operations are drawn. Those aren't CSwaps they're multiple CNOTs! Regardless, it does use more gates than necessary.).
Anyways, here's the not-quite-equivalent-but-still-a-5-qubit-code encoding circuit that I found:
And here's a quirk circuit validating that it can correct all single qubit errors:
The validation works by preparing an entangled qubit to encode, picking a random single qubit error to apply, then checking that at the end that the qubit was still correctly entangled with its partner.
Here's an even more compact and symmetrical one, using X basis controls and parity controls (though it again shuffles the corrections):