We construct two new infinite families of trace codes of dimension 2m, over the ring F _p + uF _p , with u ² = u, when p is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. By Gray mapping, we obtain two infinite families of linear p-ary codes of respective lengths (p ^m -1) ² and 2(p ^m -1) ² . When m is singly even, the first family gives five-weight codes. When m is odd and p ≡ 3 (mod 4), the first family yields p-ary two-weight codes, which are shown to be optimal by application of the Griesmer bound. The second family consists of two-weight codes that are shown to be optimal, by the Griesmer bound, whenever p = 3 and m ≥ 3, or p ≥ 5 and m ≥ 4. Applications to secret sharing schemes are given.

We construct an infinite family of two-Lee-weight codes over the ring F ₂ t uF ₂ . These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. By Gray mapping, we obtain an infinite family of abelian binary two-weight codes. They are shown to be optimal by application of the Griesmer bound. An application to secret sharing schemes is given.

In this paper, two different Gray-like maps from Z _p ^α × Z _pk ^β , where p is prime, to Z _p ⁿ , n = α t βp ^k-1 , denoted by φ and Φ, respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if C is a Z _p Z _p k-additive code, and C ^⊥ is its dual, then the weight enumerators of the image p-ary codes φ(C) and Φ(C ^⊥ ) are formally dual. This is a partial generalization of [D. S. Krotov, On Z ₂ k-dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532-1537], and the result is generalized to odd characteristic p and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed Z _p Z _p ² ... Z _p k alphabet is given.