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A generalized Forney formula for algebraic-geometric codes

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1 Author(s)
Leonard, D.A. ; Dept. of Discrete & Stat. Sci., Auburn Univ., AL, USA

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

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Information Theory, IEEE Transactions on  (Volume:42 ,  Issue: 4 )