DocumentCode
1256796
Title
Which linear codes are algebraic-geometric?
Author
Pellikan, R. ; Shen, B.-Z. ; van Wee, G.J.M.
Author_Institution
Dept. of Math. & Comput. Sci., Eindhoven Univ. of Technol., Netherlands
Volume
37
Issue
3
fYear
1991
fDate
5/1/1991 12:00:00 AM
Firstpage
583
Lastpage
602
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
encoding; error correction codes; Goppa codes; Goppa´s construction; algebraic-geometric codes; binary (7,4,3) code; divisor; infinite series of curves; linear codes; q-ary Hamming codes; rational points; redundancy; Books; Conferences; Galois fields; Geometry; Information theory; Linear code; Mathematics;
fLanguage
English
Journal_Title
Information Theory, IEEE Transactions on
Publisher
ieee
ISSN
0018-9448
Type
jour
DOI
10.1109/18.79915
Filename
79915
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