Decoding geometric Goppa codes using an extra place

Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the code length is smaller than the number of rational points on the curve, then this method can correct up to 12(d ∗ − 1)− s errors, where d∗ is the designed minimum distance of the code and s is the Clifford defect. The affine ring with respect to a place P is the set of all rational functions which have no poles except at P , and it is somehow similar to a polynomial ring. For a special kind of geometric Goppa codes, namely CΩ(D,mP ), the decoding algorithm is reduced to solving the key equation in the affine ring, which can be carried out by the subresultant sequence in the affine ring with complexity O(n3), where n is the length of codewords. Index Terms — Geometric Goppa codes, algebraic-geometric codes, decoding, affine ring, isometry, key equation. ∗This work was presented in a talk at IEEE International Symposium on Information Theory 1991, Budapest, Hungary. The first author is with Morrison-Knudsen, Integrated Software Technology Departement, Morrison-Knudsen Plaza IV-7, Boise ID 83707, USA. The last two authors are with the Eindhoven University of Technology, Department of Mathematics and Computing Science, PO Box 513, 5600 MB Eindhoven, The Netherlands.