WordNet records no synonym and no antonym for public school. It is the only member of each of its synsets, which is common for technical, taxonomic and single-sense vocabulary — there is no other English word that means the same thing. The broader and narrower terms below are the useful relations in that case.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
a tuition free school in the United States supported by taxes and controlled by a school board
private independent secondary school in Great Britain supported by endowment and tuition
Synonyms, antonyms and the broader and narrower relations for public school come from Princeton University’s WordNet. Full sources and licences.