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Which linear codes are algebraic-geometric?

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3 Author(s)
R. Pellikan ; Dept. of Math. & Comput. Sci., Eindhoven Univ. of Technol., Netherlands ; B. -Z. Shen ; G. J. M. van Wee

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 conditions are imposed on the degree of the divisor use, then criteria are derived for linear codes to be algebraic-geometric. In particular. the family of q-ary Hamming codes is investigated, and it is proven that only those with redundancy one or two and the binary (7,4,3) code are algebraic-geometric in this sense. For these codes. the authors explicitly give a curve, rational points, and a divisor. It is proven that this triple is in a certain sense unique in the case of the (7,4,3) code.

Published in:

IEEE Transactions on Information Theory  (Volume:37 ,  Issue: 3 )