| Budget Amount *help |
¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1998: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1997: ¥500,000 (Direct Cost: ¥500,000)
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| Research Abstract |
A KAPPA + 1-dimensional MDS codes of word-length n over a finite field defines an n-arc in the KAPPA-dimensional projective space over the finite field, and vice versa. Under the identification of an MDS code and an arc in this sense, their automorhism groups are isomorphic.
1. Results of our research(see the research report for details)
(1) A non-singular plane algebraic curve of degree n defined over the complex number field is called the most symmetric if the order of its projective automorphism group is not less than the order of the projective automprhism group of any non-singualr plane algebraic curve of degree n. When n = 3, 5 or 7, the most symmetric curve is projectively equivalent to the Fermat curve. When n=6, the most symmetric curve is projectively equivalent to the Wiman sextic. It has been known that the Klein quartic is the most symmetric among non-singular quartics.
(2) The set of flexes of the Klein quartic and the Wiman sextic is, respectively, a 24-arc and 72-arc, whose automorphism group is PSL(2, 7) and PSL(2, 9)=A_6 respectively.
2. Problems for further research
(1) A compact Riemann surface of genus g is called the most symmetric if the order of its holomorphic automorphism group is not less than the order of the holomorphic automorphism group of any compact Riemann surface of genus g. Is the Wiman sextic is the most symmetric among the compact Riemann surfaces of genus 10?
(2) Does a space algebraic curve gives rise to a good (MDS) code?
(3) Determine the most symmetric hyperplanes of degree n in the sense of (1) in 1.
(4) When a finite (and simple) group G is given, find G-invarinat algebraic varieties(hyper- planes etc.) and construct good (MDS) codes.
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