It's against the law to register a company name containing binary characters

This is a ridiculous name to choose. But at the same time it isn't breaking any rules (unless it's too long?) as numbers-only names are allowed (plenty of those, inluding 'binary').

Sounds like an applicant trying to be too clever for fun (and tweeting opportunities) versus a civil servant who did not understand anything the IT-minded guy in the office told them ("this is binary code for 'Don't panic'" or something like that and they thought that meant hackers were planning a terrorist attack...)

Why not just go with: 101010

And cut out the middle man?

i'm sure you would have been fine if you hadn't have made it executable.

Companies House did the right thing.