A Family of Codes on Projective Surface

1. 13th International Workshop on Algebraic and Combinatorial Coding Theory, June 15-21, Pomorie, BULGARIA A Family of Codes on Projective Surface Kouichi Shibaki (Iwate Prefectural University) Kiyoshi Nagata (Daito Bunka University)

2. Contents Brief Overview of the Algebraic Geometric Code Goppa Code General Construction of Code on Projective Scheme Code on Projective Surface Our Proposed Code on P2 Definition of the Map Definition of the Proposed Code and some Properties Hansen’s Evaluation Value for the Minimum Distance in Our Case Comparison of our Result and Hansen’s Result Conclusion

3. Brief Overview of the Algebraic Geometric Code • Codes on Polynomial ring: Reed-Solomon Codes, generalized Reed-Solomon Codes, etc. • Codes on Fractional Function Field: Classical Goppa Code, • V. D. Goppa, “A New Class of Linear Error-Correcting Codes”, Problems of Information Transmission, 6 (3), 207–212, 1970 • Codes on Algebraic Curve: Goppa Code • V. D. Goppa, “Codes on Algebraic Curves”, Soviet Math. Dokl., 24 No.1,170–172, 1981 • M. T. Tsfasman, S. G. Vladut, and T. Zink, “Modular Curves, Shimura Curves, and Goppa Codes Better than the Varshamov-Gilbert Bound”, Math. Nachr., 109, 21–28, 1982

4. Code on projective Scheme: the image of the germ map • M. T. Tsfasman and S. G. Vladut, “Algebraic Geometric Codes” , Kluwer Academic Publisher, 1991 • Code on projective Surface: • S. H. Hansen, “Error-Correcting Codes from higher-dimensional varieties”, PhD Thesis, University of Aarhus, 2001 • C. Lomont, “Error-Correcting Codes on Algebraic surfaces”, PhD Thesis, Purdue University, 2003 • P. Zampolini, “Algebraic Geometric Codes on Curves and Surfaces”, Master Program in Mathematics Faculty of Science University of Padova, Italy, 2007

5. Goppa Code • C : non-singular algebraic curve of genus g over • E : a divisor s.t.and put • Then the Goppa code is the image ofand if 2(g-1)<deg E<n, then • ΦL; injective • The dimension k=dim E=deg E-g+1 • The minimum distance δ≧n-k-g+1

6. General Construction of Code on Projective Scheme (M. T. Tsfasmann and S. G. Vladut)

7. Code on Projective Surface Note that With , where E = div( {Rλ}λ ). Z(s) =

8. Our Proposed Code on

9. Definition of the Map with where

10. Definition of the Proposed Code and some Properties

11. n : the number of point on each curve Ci m : the number of curves e : the degree of divisor E d : the degree of curves Ci (i=1,…,m)

12. Hansen’s Evaluation Value for the Minimum Distance in Our Case • Noticing thatwhere gi are polynomials of degree d defining the curves Ci, and deg(fsR)=e. • Then and the Hansen’s evaluation value for δ is

13. Comparison of our Result and Hansen’s Result • δ : our evaluation value of minimum distance • δH : Hansen’s evaluation value δ = δH +

14. Conclusion • Propose a simple construction of Hansen’s type algebraic geometric code on the Projective Plane • Calculate the dimension and the minimum distance of the proposed code • Show that our proposed code has better minimum distance than Hansen’s general type code.

15. Future Works! • Construction of Dual code • Effective decoding method

16. Благодаря!Thank You!