as can be corrected without knowing their locations. Error locator polynomial There is a linear recurrence relation that gives rise to a system of linear equations. Solving those equations identifies those error locations Xk. Define the error locator polynomial as The zeros of are the reciprocals . This follows from the above product notation construction since if then one of the multiplied terms will be zero , making the whole polynomial evaluate to zero. Let be any integer such that . Multiply both sides by and it will still be zero. Sum for k = 1 to ν and it will still be zero. Collect each term into its own sum. Extract the constant values of that are unaffected by the summation. These summations are now equivalent to the syndrome values, which we know and can substitute in! This therefore reduces to Subtracting from both sides yields Recall that j was chosen to be any integer between 1 and v inclusive, and this equivalence is true for any and all such values. Therefore, we have v linear equations, not just one. This system of linear equations can therefore be solved for the coefficients Λi of the error location polynomial: The above assumes the decoder knows the number of errors ν, but that number has not been determined yet. The PGZ decoder does not determine ν directly but rather searches for it by trying successive values. The decoder first assumes the largest value for a trial ν and sets up the linear system for that value. If the equations can be solved (i.e., the matrix determinant is nonzero), then that trial value is the number of errors. If the linear system cannot be solved, then the trial ν is reduced by one and the next smaller system is examined. Find the roots of the error locator polynomial Use the coefficients Λi found in the last step to build the error location polynomial. The roots of the error location polynomial can be found by exhaustive search. The error locators Xk are the reciprocals of those roots. The order of coefficients of the error location polynomial can be reversed, in which case the roots of that reversed polynomial are the error locators (not their reciprocals ). Chien search is an efficient implementation of this step. Calculate the error values Once the error locators Xk are known, the error values can be determined. This