Error-locator ideals for algebraic-geometric codes

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<½δ_m*, with δ_m*:=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