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Results to get maximal quasihermitian curves. New possibilities for AG codes. (English) Zbl 1072.11043
Blaum, Mario (ed.) et al., Information, coding and mathematics. Proceedings of workshop honoring Professor Bob McEliece on his 60th birthday, Pasadena, CA, USA, May 24–25, 2002. Boston, MA: Kluwer Academic Publishers (ISBN 1-4020-7079-9/hbk). The Kluwer International Series in Engineering and Computer Science 687, 55-62 (2002).
Let \(C\) be a smooth projective curve of genus \(g>0\), defined over \(\mathbb{F}_2\). In this note the authors prove that the condition \[ \sharp C(\mathbb{F}_{2^r})=2^r+1,\qquad 1\leq r\leq g, \] implies that \(C\) is maximal over \(\mathbb{F}_{2^{2g}}\). This is clear, since this condition implies immediately that the numerator of the zeta function of \(C\) is \(1+2^gx^{2g}\). Some examples of quasihermitian curves of genus \(2,\,3,\,5\) satisfying this condition are also given.
For the entire collection see [Zbl 1054.94001].
MSC:
11G20 Curves over finite and local fields
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
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