Linear codes over <inline-formula><tex-math id="M1">$ \mathbb{Z}_{p^2} $</tex-math></inline-formula> with few-weight from down-sets

Several infinite families of minimal and optimal linear codes are obtained from simplicial complexes or down-sets by Hyun and Wu et al. Inspired by their work, we present a construction of codes over $ \mathbb{Z}_{p^2} $ by employing down-sets with one maximal element or two maximal elements. We investigate five families of linear codes over $ \mathbb{Z}_{p^2} $, and determine the parameters and Homogeneous weight distributions of them. Moreover, we obtain several classes of $ p $-ary few-weight linear codes under the Gray map. Some of them are minimal or distance optimal under certain conditions.