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Then the complement Q = D − Q ∼ (n − m∗ )P∞ + E. As in the first proof, Q + ER ∼ ((a0 + deg E)(q + 1) − a1)P∞ and a1 ≤ a0 + deg E. 42. Let A = B = (a0 (q + 1) − q)P∞ , Z = (q − a0 − 1)P∞ , 1 ≤ a0 ≤ q − 1. Then the code CΩ (G, D) with G = (2a0 −1)(q +1)−a0 has d = d∗ +(q −a0 −1). This corresponds to the case G = ((q − 2)(q + 1) + (2a0 + 1 − q)(q + 1) − a0 )P∞ in the theorem above, with a0 ≥ 2a0 + 1 − q if and only if a0 ≤ q − 1. For Hermitian one-point codes of length q 3 , the q 3 finite rational points form a complete intersection with coordinate ring 2 F[x, y]/(y q + y − xq+1 , xq − x) = xi y j : 0 ≤ i ≤ q 2 − 1, 0 ≤ j ≤ q − 1 .

97, Cambridge Tracts in Mathematics, (Cambridge University Press, Cambridge, 1991). [63] H. Niederreiter and C. P. Xing, Low-discrepancy sequences obtained from algebraic function fields over finite fields, Acta Arith. 72(3), 281–298, (1995). [64] H. Niederreiter and C. Xing, Rational points on curves over finite fields: theory and applications. vol. 285, London Mathematical Society Lecture Note Series, (Cambridge University Press, Cambridge, 2001). [65] S. Park, Applications of algebraic curves to cryptography, Thesis (University of Illinois, Urbana, 2007).

In Coding theory and algebraic geometry (Luminy, 1991), vol. , pp. 18–25. Springer, Berlin, (1992). D. Ehrhard, Achieving the designed error capacity in decoding algebraicgeometric codes, IEEE Trans. Inform. Theory. 39(3), 743–751, (1993). N. D. Elkies. Explicit modular towers. In Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing (Univ. of Illinois at Urbana-Champaign. N. D. Elkies. Excellent codes from modular curves. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp.

### 4th Geometry Festival, Budapest

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