You can, perhaps
certainly it is possible...but it is not an easy prospect.
The process is straightforward. You can begin the process here
To be successful, you will need to demonstrate, under the circumstances, that an upgrade to the charachterization (or in this case, the RE code, is the appropriate result in your case. Typically this would be done by showing that the result at the time was wrong for some reason...for example if there was some error or mistake made during your out processing that caused them to assign the wrong code.
The board will also consider your post service conduct, but this to a lesser degree than than your service conduct.
Unlike the urban legend, there is no "automatic upgrade" for discharges
...in fact only a small percentage of requests are granted.
To have the greatest chance for success you should consider working with an attorney that can help you assemble the package you will submit. A google search under "military lawyer upgrade discharge" will net you several prospects...I would talk to several to see if one can be of help...but understand this is a long shot...beware of promises.
BotXXXXX XXXXXne: Its possible but difficult to do. Best of luck
Please let me know if you have more questions and I will do my best to assist, otherwise, please accept so I may get credit for my work.