Fast Computation of Gauss Sums and Resolution of the Root of Unity Ambiguity

We present an algorithm for computing Gauss sums over Fq for large prime powers q. This allows us to find the exact values of Gauss sums in many previously intractable cases. The efficient computation of such Gauss sums up to multiplication with a root of unity is achieved by using Stickelberger’s factorization of Gauss sums, an application of the Fincke-Pohst algorithm, and fast arithmetic for subfields of cyclotomic fields. This still leaves the crucial problem of resolving the root of unity ambiguity, which we settle by recursively computing “H-polynomials”. These are integer polynomials which interpolate Gauss sums at certain roots of unity. As an application, we compute the weight distribution of all binary irreducible cyclic codes of length n and dimension k with 2 k−1 n ≤ 5, 000.