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A converter box is essentially a cable box, so using apex tv codes will not work, you'll need to use a cable box or satellite receiver type code associated with apex equipment.
2478 should be the correct code for your converter.
This is, however, assuming your particular remote is compatible with the apex converter box. Universal remotes are typically limited in their compatibility, especially with less common products and brands (such as apex, and converter boxes).
Your best option may be to purchase a true universal remote to control all of your devices, such as a logitech harmony (click here for example) that will work for all of your devices. It is programmed through an easy to use software program on your computer, that downloads the most up to date remote and device codes from the internet, so it is virtually guaranteed to work with all equipment. It will also give you easy to use macro functionality, so you can press a single button to watch a movie for example, and it will turn on the dvd player, tv, and surround system, tune to the correct inputs on the tv, and start the movie playing, etc. Because of the online database with specific codes for each model, this type of remote has much greater functionality than the basic universal remotes that you program based on a book of outdated manufacturer codes.
this did not work.
As I mentioned, that is the specific code for your dynex converter. If it isn't working, it is a limitation of your radio shack remote, not a problem with your box or the code. Again, in this case, your only real option to get a remote working with the box will be to purchase a higher end universal remote that accepts a wider range of programming, such as the harmony models I linked to earlier.
As an alternative, it may be cheaper for you to just by a new converter box. You can get a new one for $30-40 which is less than a high end remote would be.