Two "dual" families of Nearly-Linear Codes over ℤ p , p odd

Since the paper by Hammons e.a. [1], various authors have shown an enormous interest in linear codes over the ring Z4. A special weight function on Z4 was introduced and by means of the so called Gray map ¿ : Z4¿Z2 2 a relation was established between linear codes over Z4 and certain interesting non-linear binary codes of even length. Here, we shall generalize these notions to codes over Z p2 where p is an arbitrary prime. To this end, a new weight function will be proposed for Z p2 . Further, properties of linear codes over Z p2 will be discussed and the mapping ¿ will be generalized to an isometry between Z p2 and Z p p , resp. between Z p2 n and Z p pn . Some properties of Galois rings over Z q will be described and two dual families of linear codes of length n = p m - 1, gcd(m, p) = 1, over Z q will be constructed. Taking q = p 2, their images under the new mapping can be viewed as a generalization of the binary Kerdock and the Preparata code, although they miss some of their nice combinatorial properties.