Hi there and welcome.
You need to stop the network adapter from sleeping
To do this do the following please
Click start, right click computer and select manage
Go to Device Manager, right-click the network adapter(that keeps getting turned off), and select Properties.
Save, reboot and it should not disconnect the xbox anymore.
Please let me know if you still have problems
ok, then I want to make sure the latest firmware is on the livebox please
Can you go to the below link and follow the instructions to update it
If you have any difficulties, please do not hesitate to contact me