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abelian Advanced Chapters algebraic curves algebraic geometry bound algebraic geometry codes algorithm arbitrary automorphism BCH codes called code of genus codes with parameters codeword coding theory Consider construction coordinates COROLLARY corresponding curve of genus decoding defined definition degD denoted differential form dimension Div(X divisors of degree effective divisors elements elliptic curves embedding equation equivalent example EXERCISE exists F9-points finite field formula function field genus g Gilbert-Varshamov bound given Hence Hint hyperplane inequality integer irreducible isomorphic Lemma Let q line bundle linear code lower bound map f matrix maximal minimum distance morphism nonlinear codes nonzero number of points obtain P C X poles polynomial prime projective systems PROOF PROPOSITION rational function rational map rational points Reed-Solomon code regular map REMARK Riemann Riemann-Roch theorem ring self-dual codes Show space subset subspace upper bound variety vector weight zero zeta function
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