Abstract
In this paper, we study self-dual codes over finite fields using tools from algebraic function fields in one variable. An algebraic geometry code of length n is defined using two divisors G and D = P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> + ⋯ + P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> . We characterize self-orthogonality of the genus zero code in terms of the divisors G, D and the value of a well-chosen derivative polynomial at points (P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1≤i≤n</sub> . We explore the existence problem of MDS self-dual codes in the odd characteristic case, and we explicitly construct families of new MDS self-dual codes.
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