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.

A q-ary code of length n, size M, and minimum distanced is called an (n,M,d)q code. An (n,q ^k ,n - k + 1) _q code is called a maximum distance separable (MDS) code. In this paper, some MDS codes over small alphabets are classified. It is shown that every (k + d - 1, q ^k , d) _q code with k ≥ 3, d ≥ 3, q ∈ (5, 7} is equivalent to a linear code with the same parameters. This implies that the (6, 5 ⁴ , 3) ₅ code and the (n, 7 ^n-2 , 3) ₇ MDS codes for n ∈ (6, 7, 8} are unique. The classification of one-error-correcting 8-ary MDS codes is also finished; there are 14, 8, 4, and 4 equivalence classes of (n, 8 ^n-2 , 3) ₈ codes for n = 6, 7, 8, and 9, respectively. One of the equivalence classes of perfect (9, 8 ⁷ , 3) ₈ codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.

We characterize the set of all n-ary quasigroups of order 4: every n-ary quasigroup of order 4 is permutably reducible or semilinear. Permutable reducibility means that an n-ary quasigroup can be represented as a composition of k-ary and $(n-k+1)$-ary quasigroups for some k from 2 to $n-1$, where the order of arguments in the representation can differ from the original order. The set of semilinear n-ary quasigroups has a characterization in terms of Boolean functions.