13th International Workshop on Algebraic and Combinatorial Coding Theory, June 15-21, Pomorie, BULGARIA

A Family of Codes on A Family of Codes on Projective SurfaceProjective Surface

A Family of Codes on A Family of Codes on Projective SurfaceProjective Surface

Kouichi Shibaki (Iwate Prefectural University)Kiyoshi Nagata (Daito Bunka University)

ContentsContentsContentsContents 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

Brief Overview of Brief Overview of the Algebraic Geometric Codethe Algebraic Geometric CodeBrief Overview of Brief Overview of the Algebraic Geometric Codethe 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

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

C : non-singular algebraic curve of genus g over E : a divisor s.t.

and put Then the Goppa code is the image of

and 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

Goppa CodeGoppa CodeGoppa CodeGoppa Code

General Construction of Code on General Construction of Code on Projective SchemeProjective SchemeGeneral Construction of Code on General Construction of Code on Projective SchemeProjective Scheme

(M. T. Tsfasmann and S. G. Vladut)

Code on Projective SurfaceCode on Projective SurfaceCode on Projective SurfaceCode on Projective Surface

Note thatWith

, where E = div( {Rλ}λ ).

Z(s) =

Our Proposed Code onOur Proposed Code onOur Proposed Code onOur Proposed Code on

Definition of the MapDefinition of the MapDefinition of the MapDefinition of the Map

with

where

Definition of the Proposed Code and Definition of the Proposed Code and some Propertiessome PropertiesDefinition of the Proposed Code and Definition of the Proposed Code and some Propertiessome Properties

n : the number of point on each curve Ci m : the number of curvese : the degree of divisor Ed : the degree of curves Ci (i=1,…,m)

Hansen’s Evaluation Value for the Hansen’s Evaluation Value for the Minimum Distance in Our CaseMinimum Distance in Our CaseHansen’s Evaluation Value for the Hansen’s Evaluation Value for the Minimum Distance in Our CaseMinimum Distance in Our Case Noticing that

where gi are polynomials of degree d defining the curves Ci, and deg(fsR)=e.

Then and the Hansen’s evaluation value for δ is

Comparison of our Result and Comparison of our Result and Hansen’s ResultHansen’s ResultComparison of our Result and Comparison of our Result and Hansen’s ResultHansen’s Result

δ : our evaluation value of minimum distance δH : Hansen’s evaluation value

δ = δH +

ConclusionConclusionConclusionConclusion

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.

Future Works!Future Works!Future Works!Future Works!

Construction of Dual codeEffective decoding method

БлагодаряБлагодаря!!

Thank You!Thank You!

БлагодаряБлагодаря!!

Thank You!Thank You!