What distribution and kernel version do you use? You can set the value in /proc/sys/kernel/core_pattern if you have a recent enough kernel. You can tell the current value like this:
$ cat /proc/sys/kernel/core_pattern
The output there, using apport is typical on Debian based systems, and is used for the crash handler. Assuming you don't mind losing that functionality, you can just set the value like this (on systems supporting sudo - use root user if your system does not support sudo):
$ sudo bash -c 'echo "/home/core/core.%u.%p" > /proc/sys/kernel/core_pattern'
There may well be a way to customize the name of the file using apport directly - I am unfamiliar with it, so I'm just telling you how to do it in a way I know.
Anyhow, once you've set that, provided your user's limits are set such that core files are created, core dumps will be put in the /home/core directory with the name "core.%u.%p", where the %u is the UID of the user whose process was dumped, and %p is the PID of the process. For a list of all the different % substitutions, see the core