I want to install Ubuntu 16.04 in my laptop, it is a Asus x555UB, I have already installed Windows 10(It's the default OS of the laptop, it came preinstalled, I disabled fast boot and secure boot), I don't want to uninstall Windows, I want to have both OS. But when I am going to install Ubuntu and when the partition screen appear this message is displayed:

ubi-partman failed with exit code 10. Further information may be found in /var/log/syslog

I reviewed the syslog and the problem was that I don't have enough space, but I have enough space so I was reading a lot about this and I found that the cause is I have AHCI mode active in my BIOS, and the solution that I found is disable it, but when I was to the BIOS interface I can't change the AHCI mode, so it's possible to install Ubuntu with the AHCI mode active?

Yes, it is possible.

See this link -->http://wiki.archlinux.org/index.php/AHCI

I do not run Ubuntu, so I can not help there. Have a look at the Ubuntu forums, there should be people there that know how to make an install in AHCI mode.

Use the Windows disk manager to reduce the size of a Windows "drive" to create unused (unallocated) space.
Here is a guide (written for Mint) which will give you the basic procedure.

http://forums.linuxmint.com/viewtopic.php?p=912036