The best known constructions for arrays with low bias are those from [1] and the exponential sum method based on the Weil-Carlitz-Uchiyama bound. They all yield essentially the same parameters. We present new efficient coding-theoretic constructions, which allow far-reaching generalizations and improvements. The classical constructions can be described as making use of Reed-Solomon codes. Our recursive construction yields greatly improved parameters even when applied to Reed-Solomon codes. Use of algebraic-geometric codes leads to even better results, which are optimal in an asymptotic sense. The applications comprise universal hashing, authentication, resilient functions and pseudorandomness.