Which linear codes are algebraic-geometric?

An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If one imposes conditions on the degree of the divisor used, then we derive criteria for linear codes to be algebraic-geometric. In particular, we investigate the family of q-ary Hamming codes, and prove that only those with redundancy one or two, and the binary [7, 4, 3] code are algebraic-geometric in this sense. For these codes we explicitly give a curve, rational points and a divisor. We prove that this triple is in a certain sense unique in the case of the [7, 4, 3] code.