Projective geometrical study of MDS codes
Project/Area Number  09640270 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Osaka Prefecture University 
Principal Investigator 
KANETA Hitoshi College of Engineering, Osaka Prefecture University, Professor, 工学部, 教授 (10093014)

CoInvestigator(Kenkyūbuntansha) 
島川 和久 岡山大学, 理学部, 助教授 (70109081)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

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)

Keywords  MDS code / finite geometry / arc / algebraic curve / automorphism group / flex / MDS符号 / 有限幾何 / アーク / 代数曲線 / 自己同型群 / 変曲点 / 6次曲線 / 対称的な曲線 
Research Abstract 
A KAPPA + 1dimensional MDS codes of wordlength n over a finite field defines an narc in the KAPPAdimensional 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 nonsingular 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 nonsingualr 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 nonsingular quartics. (2) The set of flexes of the Klein quartic and the Wiman sextic is, respectively, a 24arc and 72arc, 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 Ginvarinat algebraic varieties(hyper planes etc.) and construct good (MDS) codes.

Report
(4results)
Research Output
(5results)