I have now successfully installed Ubuntu!
I received further feedback from ASUS telling me that the X502A
does allow you to disable Secure Boot and enable lagacy boot mode. I originally contacted them as I couldn't work out where one the BIOS options was; the subsequent e-mail exchange seems to have got a bit confused because, at least in hindsight, the solution is pretty obvious. I had to disable Secure Boot (the option I looked through initially) and Fast Startup and then restart the computer. I was then able to enable the legacy boot mode and launch the Ubuntu installer.
Thanks, Andy —
Andy Sep 22 '13 at 18:26