Decoding Of A Class Of Algebraic Geometric Codes

ABSTRACTThe thesis is devoted to the decoding of a class of algebraic-geometric codes. After some conception and properties on linear block codes. algebraic curves and algebraic-geometric codes introduced, the conception of recurring relation on sequence is investigated, and the concept of A-type recurring relation is proposed. Then the agreement theorem and generalized Berlekamp-Massev algorithm as well as majority voting scheme are established. An A-type recurring relation is introduced on the syndrome sequence of an error vector. Lastly an efficient decoding algorithm is presented for one point algebraic-geometric codes. Main contributions of this work are as follows:(1)We generalize the concept of linear recurring relation and propose a new type of recurring relation on sequence-type recurring relation on sequence. This type of recurring relation depends on a group of given polynomials, and in general is non-linear.(2)We propose the concepts of minimal polynomial set of A-type of recurring relation and d-set which describes the length of .A-type recurring relation.(3)We establish some essential properties such as the agreement theorem etc. for A-type recurring relation satisfying some conditions.(4)By generalizing Berlekamp-Massey algorithm we develop a generalized Berlekamp-Massey algorithm for A-type recurring relation satisfying some conditions and discuss its complexity. Using the algorithm we can calculate a minimal polynom~al set of A-type recurring relation.(5)We develop a majority voting scheme for A-type recurring relation.(6)For one point algebraic-geometric codes we introduce an A-type recurring relation on the syndrome sequence of an error vector. And we show that the size of d-sets of the A-type recurring relation is limited by the number of errors by using knowledge of algebraic curves.(7)Using the generalized Berlekamp-Massey algorithm and majority voting scheme we present an efficient decoding algorithm for one point algebraic-geometric codes, whose complexity achieves the most advanced world standards in the same class of algorithms.