Locally repairable codes and minimal codes by homogenization of down-sets

We construct some families of p-ary minimal and distance-optimal codes and some families of locally repairable p-ary codes by the homogenization method for a prime p. For this, we use the code $$C_{(D_{F_h})^c}$$ associated with the complement of the defining set $$D_{F_h}$$ generated from the homogenization $$F_h$$ of a multi-variable function F. We first find two criteria: one is a criterion for the code $$C_{(D_{F_h})^c}$$ to be a minimal code, and the other is a criterion for $$C_{(D_{F_h})^c}$$ to be an LRC (Locally Repairable Code) with locality $$t\ge 2$$ . Then we focus on the codes $$C_{(\Delta _h)^{c^*}}$$ , which are the cases where the defining sets $$D_F$$ are certain down-sets $$\Delta$$ of $$\mathbb {F}_p^n$$ generated by one maximal element with support size at most two. We obtain several infinite families of minimal p-ary linear codes, using the first criterion; some of them are also distance-optimal. Furthermore, we produce several infinite families of LRCs with locality two including an infinite family of alphabet-optimal LRCs, using the second criterion.