Which linear codes are algebraic-geometric?
- 1 May 1991
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 37 (3) , 583-602
- https://doi.org/10.1109/18.79915
Abstract
An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If conditions are imposed on the degree of the divisor use, then criteria are derived for linear codes to be algebraic-geometric. In particular. the family of q-ary Hamming codes is investigated, and it is proven that only those with redundancy one or two and the binary (7,4,3) code are algebraic-geometric in this sense. For these codes. the authors explicitly give a curve, rational points, and a divisor. It is proven that this triple is in a certain sense unique in the case of the (7,4,3) code.Keywords
This publication has 11 references indexed in Scilit:
- Algebraic-Geometric CodesPublished by Springer Nature ,1991
- Group codes on certain algebraic curves with many rational pointsApplicable Algebra in Engineering, Communication and Computing, 1990
- Introduction to Coding Theory and Algebraic GeometryPublished by Springer Nature ,1988
- Algebraic Groups and Class FieldsPublished by Springer Nature ,1988
- Geometry of Algebraic CurvesPublished by Springer Nature ,1985
- Codes and informationRussian Mathematical Surveys, 1984
- ALGEBRAICO-GEOMETRIC CODESMathematics of the USSR-Izvestiya, 1983
- Introduction to Coding TheoryPublished by Springer Nature ,1982
- On generalized ReedMuller codes and their relativesInformation and Control, 1970
- Introduction to the Theory of Algebraic Functions of One VariablePublished by American Mathematical Society (AMS) ,1951