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Error-locator ideals for algebraic-geometric codes

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

The error locations for an algebraic-geometric code C*(D,mP) are exactly the common zeros (that is, a projective variety V(I)) of a set (ideal) I of error-locator functions. The paper gives a one-dimensional Berlekamp-Massey version of the Feng-Rao (1993) algorithm for decoding algebraic-geometric codes C*(D,mP). This produces a generating set for I (as an ideal) of size at most ρ (the smallest positive pole order at P of any function in L(mP)) relative to any error of weight at most e<½δm*, with δm*:=m-2g+2 the designed minimum distance of the code. This algorithm requires at most c(ρm2+Nρm+ρ2m) field multiplications, with c a small constant, and N a small constant function of the curve. The error-positions are then given as exactly the common zeros of generator functions of the error-locator ideal I

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Information Theory, IEEE Transactions on  (Volume:41 ,  Issue: 3 )