Binary constant weight codes have important applications and have been studied for many years. Optimal or near-optimal binary constant weight codes of small lengths have been determined. In this paper we propose a new construction of explicit binary constant weight codes from $q$-ary Reed-Solomon codes. Some of our binary constant weight codes are optimal or new. In particular new binary constant weight codes $A(64, 10, 8) \geq 4108$ and $A(64, 12, 8) \geq 522$ are constructed. We also give explicitly constructed binary constant weight codes which improve Gilbert and Graham-Sloane lower bounds in some range of parameters. An extension to algebraic geometric codes is also presented.