Two “Dual” Families of Nearly-Linear Codes over ℤ _p , p odd

Abstract.

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 ϕ : 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 ϕ 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.