  We construct families of locally recoverable codes with availability $t\geq
2$ using fiber products of curves, determine the exact minimum distance of many
families, and prove a general theorem for minimum distance of such codes. The
paper concludes with an exploration of parameters of codes from these families
and the fiber product construction more generally. We show that fiber product
codes can achieve arbitrarily large rate and arbitrarily small relative defect,
and compare to known bounds and important constructions from the literature.
