Abstract

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. Key words: algebraic-geometric codes, algebraic curves, divisors, generalized Goppa codes, geometric Goppa codes. I. Introduction Since the early papers by Goppa [5],[6],[7], [8], algebraic-geometric codes have been in the spotlight of coding theoretic research for about a decade. As is well-known, numerous exciting results have been achieved using Goppa's construction of li...

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