Best way to reduce file size keeping Quality

You should contact whoever is authoring and mater the disks. They probably have specific settings that are required for that to work.

In general, you should never export a file from FCP, always export a reference file and encode in compressor or another compression program.

 

Use Compressor to specify a target bitrate of 1400Kbps for video and 192Kbps for audio.

 

There are Bitrate calculators available online, like this one: http://www.videohelp.com/calc.htm
They help you to get the right size.

The bitrate is simply the number of bits and bytes that are transferred per second. The more of them bits are transferred, the better is your quality, but the file size is bigger. 1400 is a good, well somewhat, compromise between size and quality. 192Kbps is a standard.

A reference movie is actually only a kind of database that tells Compressor or QT where to find the specific clip and what you did to it (filters, color corrections, transitions). There is no re-compression (possible generation loss!) taking place on output, and you save a lot of space.
There is one caveat though: You should only use it when your project with all its media stays at the same place on the same computer. If you want to share your movie with someone else, it won't work.

 

Bitrate is file size over time (akin to miles per hour). 1600Kbps for 10 mins = 120MB. 1400 and 192 is a good balance of video and audio quality.

 

You will probably want to use h264 encoding which is one of the most efficient. This can be both a MP4 or MOV file. You always loose quality when you decrease size so you have to tweak your settings to see what's acceptable for you.