Higher Power Residue Codes

For certain primeslandp, and characters χ:F*p→F*l2, we construct codesWof lengthp+ 1 overFl2which are linear overFl, but not overFl2, and which are invariant under a monomial action of the group SL(2,p). We consider the cases of cubic and quartic characters in detail and use theWto construct linear codes overFlin these cases.