Summary of decoding algebraic geometry codes

Abstract

P - QS = 0 mod g. 3) Evaluate the residues of Gdz. For non-zero genus the same general program is possible, but enormous difficulties must be overcome. The first step is the description of a new finite data structure for the affine coordinate ring, permitting efficient arithmetic algorithms. To define the data structure, existence of a non-singular affine embedding with one point at infinity must be proved. . Syndrome construction and conversion of the decoding problem to a polynomial congruence is made possible by the definition of a polynomial whose divisor of zeros is exactly G and by the construction of a space of differentials which corresponds to F; and contains Q(G - D). Euclid’s algorithm is generalized for non-Euclidean affine coordinate rings. First, a generalized resultant matrix is defined. By selecting pivots in a certain order during row-reduction on the resultant matrix, objects much like remainders and convergents occur. It is proved that these objects provide minimal solutions to the polynomial congruence. 0 mod G need not be a minimal solution of 5 - S 3 0 mod G. The % error loss stems from this. Further degradation of decoding capacity occurs for some choices of the curve X, the point P, and the divisor S. The algorithm consists of efficient, easily implemented matrix operations. However, the algebraic geometry and approximation techniques necessary to develop and understand the method are both original and deep. errors in genus p.