Cyclic codes over GR(4/sup m/) which are also cyclic over (Zopf)/sub 4/

Let GR(4/sup m/) be the Galois ring of characteristic 4 and cardinality 4/sup m/, and (alpha)_={(alpha)/sub 0/,(alpha)/sub 1/,...,(alpha)/sub m-1/} be a basis of GR(4/sup m/) over (Zopf)/sub 4/ when we regard GR(4/sup m/) as a free (Zopf)/sub 4/-module of rank m. Define the map d/sub (alpha)_/ from GR(4/sup m/)[z]/(z/sup n/-1) into (Zopf)/sub 4/[z]/(z/sup mn/-1) by d(alpha)_(a(z))=(sigma)/sub i=0//sup m1/(sigma)/sub j=0//sup n-1/a/sub ij/z/sup mj+i/ where a(z)=(sigma)/sub j=0//sup n1/a/sub j/z/sup j/ and a/sub j/=(sigma)/sub i=0//sup m-1/a/sub ij/(alpha)/sub i/, a/sub ij/(isin)(Zopf)/sub 4/. Then, for any linear code C of length n over GR(4/sup m/), its image d/sub (alpha)_/(C) is a (Zopf)/sub 4/-linear code of length mn. In this article, for n and m being odd integers, it is determined all pairs ((alpha)_,C) such that d/sub (alpha)_/(C) is (Zopf)/sub 4/-cyclic, where (alpha)_ is a basis of GR(4/sup m/) over (Zopf)/sub 4/, and C is a cyclic code of length n over GR(4/sup m/).