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The minimum distance of some families of expander codes is studied, as well as some related families of codes defined on bipartite graphs. The weight spectrum and the minimum distance of a random ensemble of such codes are computed and it is shown that it sometimes meets the Gilbert-Varshamov (GV) bound. A lower bound on the minimum distances of constructive families of expander codes is derived. The relative minimum distance of the expander code is shown to exceed the product bound, i.e., the quantity δ0δ1 where δ0 and δ1 are the minimum relative distances of the constituent codes. As a consequence of this, a polynomially constructible family of expander codes is obtained whose relative distance exceeds the Zyablov bound on the distance of serial concatenations.