MacWilliams Equivalence Theorem for the Lee Weight over ℤ 4 1

For codes over fields, the MacWilliams equivalence theorem give us a complete characterization when two codes are equivalent. Considering the important role of the Lee weight in coding theory, one would like to have a similar results for codes over integer residue rings equipped with the Lee weight. We would like to prove that the linear isomorphisms between two codes in Z n that is preserving the Lee weight are exactly the maps of the form 1 2 1 (1) 2 (2) ( ) ( , , ) ( , , , ) , m m m f x x u x u x x u x where 1, , { 1,1} m u u and n S . The problem is still largely open. Wood proved the result for codes over Z n where n is a power of 2 or 3. In this paper we prove the result for prime n of the form 4 1 p where p is prime. (