戒田高康 (Takayasu KAIDA) 八代工業高等専門学校 情報電子工学科

1. 高次の座を用いた代数幾何符号の構成On Construction of Algebraic Geometric Codes with High Degree Places(Update : 2001.02.02) 戒田高康　(Takayasu KAIDA) 八代工業高等専門学校 情報電子工学科 Dept. of Information and Electronic Engineering, Yatsushiro National College of Technology

2. Outline of the Presentation • Background and preparations • Code with high degree places by Xing, et.al. • Function type code and residue type code with high degree places • Relation between Xing’s code and proposed code • Decoding for proposed code • An example over an elliptic function field • Conclusion and future works

3. Background • Algebraic geometric (AG) code • Its code length is restricted by Hasse-Weil bound • AG code with high degree places • C.Xing, H.Neidereiter and Y.K.Lam[2,3], 1999. • T.Kaida, K.Imamura and T.Moriuchi[5,6], Not only function type but also residue type, 1995~2000 • Relation of codes proposed in [1,2,3] • One construction is new and interested[4], 1999

4. Preparations [7]H.Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, 1993

5. Code Proposed by Xing, et.al.

6. Parameters of the code

7. Function Type Code

8. Function Type Code

9. Residue Type Code

10. Residue Type Code

11. Duality of Function and Residue Code

12. Duality of Function and Residue Code

13. Duality of Function and Residue Code

14. Duality of Function and Residue Code

15. Relation of CX and CL

16. Decoding for Proposed Codes • By generator matrix of the dual code • In extension field by

17. An Example

18. An Example (Ex. 8) Table 1. Places over F/K with deg≦3and local parameters (LP) over F, K(x) and K(y)

19. An Example

20. An Example (Ex.8) Table 2. Places over F’/K’ with deg=1and their LP

21. An Example

22. Conclusion • The definition of codes with high degree places • by Xing, et.al. (function type) • function type and residue type • The duality of function and residue type code • Function type is a special case of Xing’s code • An example over an elliptic function field

23. Future Works • Residue type of Xing’s code • Theoretical evaluation for the minimum distance of proposed code • Decoding method of proposed code • Relation between proposed code and conventional codes (AG code, sub-field sub-code, concatenated code)

24. References [1]C.Xing, H.Niederreiter and K.Y.Lam, “Constructions of algebraic-geometry codes”, IEEE Trans. Information Theory, vol.45, pp.1186-1193, May 1999. [2] H.Niederreiter, C.Xing and K.Y.Lam, “A new construction of algebraic-geometry codes”, Applicable Algebra in Engineering, Communication and Computing, vol.9, pp.373-381, Springer-Verlag, 1999. [3] C.Xing, H.Niederreiter and K.Y.Lam, “A generalization of algebraic-geometry codes”, IEEE Trans. Information Theory, vol.45, pp.2498-2501, Nov. 1999. [4]F.Ozbudak and H.Stichtenoth, “Constructing codes from algebraic curves”, IEEE Trans. Information Theory, vol.45, pp.2502-2505, Nov. 1999. [5]戒田高康, 今村恭己, 森内勉, “高次の座を用いた代数幾何符号に関する考察”, 第18回情報理論とその応用シンポジウム予稿集, pp.231-234, 花巻, 1995年10月 [6]T.Kaida and K. Imamura, “Residue type of algebraic geometric codes with high degree places”, Proc. Of International Symposium on Information Theory and Its Applications, pp.453-456, Honolulu, Nov. 2000. [7]H.Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, 1993.