Algebraic Decoding of Cyclic Codes Without Error-Locator Polynomials

The algebraic decoding of a p-ary cyclic code consists of four steps: 1) computation of the known syndromes using the received word; 2) computation of the unknown syndromes from the known syndromes; 3) computation of the error positions by a use of the Berlekamp-Massey (BM) algorithm and Chien's search; and 4) computation of the error values by solving a linear system. This paper addresses two problems of determining the error positions and computing the unknown syndromes. To solve the first problem, a new matrix, together with Gaussian elimination instead of the BM algorithm and Chien's search, is proposed. In this new simplified decoder, finding an error-locator polynomial is completely avoided. A main advantage of the presented decoding method is when the Bose-Chaudhuri-Hocquenghem bound is unequal to the minimum distance of the code. Some cyclic codes, which do not have 2t consecutive known syndromes, can be decoded up to their actual minimum distance by using the presented matrix once. To solve the second problem, two algorithms for different square matrices reported recently by Lee et al. are also provided to calculate the value of an unknown syndrome. Finally, an algebraic decoding of the (47,23,15) ternary cyclic code is given.