Finding BCH error locator polynomials in one step

Abstract

Syndrome polynomials produced in a BCH decoder span an ideal in a multivariate polynomial ring. Because the error locators satisfy every member of the ideal, polynomial ideal theory is used to find another, more easily solvable system of equations having the same roots. One such equation is the (univariate) error locator polynomial. The Letter shows how to write this polynomial in one mathematical step from the syndrome polynomials.