We study the minimum distance of the binary expansion of high-rate Reed-Solomon (RS) codes and product codes in the polynomial basis and show hat the binary codes obtained in this way usually have minimum distance equal to the designed symbol minimum distance. We then show that a judicious choice for the code roots may yield binary expansions with larger binary minimum distance and better asymptotic performance. This result is used to design high-rate RS product codes with significantly lower error floors compared to classical constructions.