We study codes with parameters of$q$-ary shortened Hamming codes, i.e.,$(n=(q^{m}-q)/(q-1), q^{n-m}, 3)_{q}$. Firstly, we prove the fact mentioned in 1998 by Brouwer et al. that such codes are optimal, generalizing it to a bound for multifold packings of radius-1 balls, with a corollary for multiple coverings. In particular, we show that the punctured Hamming code is an optimal$q$-fold packing with minimum distance 2. Secondly, for every admissible length starting from$n=20$, we show the existence of 4-ary codes with parameters of shortened 1-perfect codes that cannot be obtained by shortening a 1-perfect code.

We investigate fat projective linear codes over${\mathbb Z}_{p^{m}}$,$m\geqslant 2$, with two nonzero homogeneous weights (“two-weight codes”), building on the graph theory approach developed by Delsarte for codes over fields. Our main result is the classification of such codes under the additional assumption that the columns of a generator matrix of the code determine a cap in the projective Hjelmslev geometry$\mathop {\mathrm {PHG}}\nolimits (k-1, {\mathbb Z}_{p^{m}})$. This generalizes a result on projective two weight codes with dual distance at least four (Calderbank, 1982). The proof relies on a careful analysis of a certain strongly regular graph built on the cosets of the dual code, and on an interpretation of its parameters in terms of projective Hjelmslev geometry.

In this paper, two different Gray-like maps from Zpα× Zpkβ, where p is prime, to Zpn, n = α t βpk-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 ZpZpk-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 Z2k-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 ZpZp2... Zpk alphabet is given.