Faster solution of the key equation for decoding BCH error-correcting codes

Abstract

Over any field of any characteristic c, the coefficients of a polynomial of a degree n can be expressed via the power sums of its zeros by means of the so called key equation for decoding the BCH error-correcting codes. Berlekamp's algorithm of 1968 solves this equation by using order of n2 field operations. The subsequent algorithms of 1975-1980 use O(n(log n)2 log log n) field operations, though a considerable overhead constant is hidden in the `O' notation. Our algorithms simplify the solution and further decrease the latter asymptotic bound, by factors ranging from order of log n, for c = 0 and c>n (in which case the overhead constant drops dramatically, thus implying substantial practical improvement of the known algorithms), to order of min (c, log n), for 2≤c≤n; we use Las Vegas randomization in the latter case. We also show some applications to computing the characteristic polynomial of a matrix.