There are not custom ROMs designed to our device on the web. But its brothers Galaxy Young and Galaxy Young Duos have nice custom ROMs, and they can be used in S6313T. The only problem is: we don't have TV!
But now this problem is over. Just flash this patch using a custom recovery and enjoy your mobile TV!
EnableTV_v1.0 - Standard patch, installs necessary files and S6313T stock kernel to TV work
EnableTV_v1.0_nokernelchange - Use this ONLY if you are using a custom S6313T kernel (I don't know any btw). It installs necessary files, but doesn't touch in kernel
Galaxy Young Duos TV, Device Specific App for the Samsung Galaxy Young S6310
Current Stable Version: 1.0
Stable Release Date: 2015-11-17
Last Updated 2015-11-17