A generalized Forney formula for algebraic-geometric codes

Author

Leonard, Douglas A.

Author_Institution

Dept. of Discrete & Stat. Sci., Auburn Univ., AL, USA

Volume

42

Issue

4

fYear

1996

fDate

7/1/1996 12:00:00 AM

Firstpage

1263

Lastpage

1268

Abstract

This correspondence contains a straightforward generalization of decoding of BCH codes to the decoding of algebraic-geometric codes, couched in terms of varieties, ideals, and Grobner bases. This consists of 1) a Berlekamp-Massey-type lattice-shifting row-reduction algorithm with majority voting similar to algorithms in the current literature, 2) a realization that it produces a minimal Grobner basis B for the error-locator ideal I(V) relative to a particular weighted total degree monomial ordering, 3) a factoring of that basis into several minimal PLEX bases, that facilitates finding the variety V of error positions, and 4) a direct generalization of Forney´s formula to calculate error magnitudes using functions σp, which are by-products of this factoring

Keywords

BCH codes; algebraic geometric codes; decoding; Berlekamp-Massey-type algorithm; algebraic-geometric codes; decoding; error magnitudes; error-locator ideal; factoring; generalized Forney formula; lattice-shifting row-reduction algorithm; majority voting; minimal Grobner basis; minimal PLEX bases; weighted total degree monomial ordering; Decoding; Galois fields; Packaging; Polynomials; Software packages; Voting;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/18.508855

Filename

508855