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_(DFₕ)ᶜ associated with the complement of the defining set DFₕ generated from the homogenization Fₕ of a multi-variable function F. We first find two criteria: one is a criterion for the code C_(DFₕ)ᶜ to be a minimal code, and the other is a criterion for C_(DFₕ)ᶜ to be an LRC (Locally Repairable Code) with locality t≥ 2 . Then we focus on the codes C_(Δ ₕ)c^* , which are the cases where the defining sets DF are certain down-sets Δ of Fₚⁿ 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.