This is not an immigration quesiton.
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This question was mis-classified because you used the words re-entry permit ... I'll remove those words and you might be able to get a military/JAG lawyer to answer.
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RE4 code normally means ineligible for reenlistment. There may be waivers or appeals to change RE code but in a drawdown you should not be too optimistic in getting back in when there are plenty of recruits without your baggage..
I buy you might have received an Expert Marksman Award because you might have shot well in basic, but I don't buy that you received the Good Conduct Medal. Normally takes three three years or one year in combat. Some one who had an involuntary, entry level discharge for performance and conduct is not going to get a GCM...
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Only a recruiter can tell you whether you can reenlist, but with an RE4, the odds are next to zero. There are ways to get an RE code changed, but you have to prove that they gave you the wrong RE Code.
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I take it you were a Marine. In order to have the characterization of discharge corrected or upgraded or the RE Code changed, the only option you have it to apply to the Board for Correction of Naval Records in Washington D.C. I currently have a case pending before the Board and am very familiar with the process. Feel free to give me a call to discuss further. The initial consultation is always free. You can find out more information at the following website: http://www.donhq.navy.mil/bcnr/bcnr.htm