Jacobi sums over finite fields

Abstract

Jacobi sums over finite fields byPaul van Wamelen (Baton Rouge, LA)1. Definitions and notation. Let e be a positive integer, e > 2, and fix ζ e , a primitive eth root of unity. Let K = Q(ζ e ). Let p be a prime not dividing e and r an integer such that p r ≡ 1 mod e. Let r 0 be the least positive integer such that p r 0 ≡ 1 mod e. Note that r 0 | r. Let F q be the finite field with q = p r elements. Let γ be a generator of the cyclic group F * q . Define a multiplicative character χ e = χ : F * q → Q(ζ e ) by χ(γ) = ζ e . We extend it by χ(0) = 0 to a map from F q to Q(ζ e ). For two integers m and n the Jacobi sum J(χ m , χ n ) is defined by

New Hadamard matrices of order 4p2 obtained from Jacobi sums of order 16

“…We also list the corresponding values a, b, c, d and the choice of the generator g which gives the corresponding difference family according to Theorems 16 and 18. The values a, b, c, d were obtained with the help of Paul van Wamelen's PARI-implementation [1,5] for the computation of Jacobi sums.…”

“…We believe that, for any large enough n, our constructions yield at least 5 8 n 2 5 primes q < n, q ≡ 7 mod 16 such that a regular Hadamard matrices of order 4q 2 exists. Our approach is based on 16th and (q + 1)th cyclotomic classes.…”

Fast computation of Gauss sums and resolution of the root of unity ambiguity

“…Similarly to [29], we will view the ideal G(χ P )A as a lattice. Here, by a lattice we mean a free Z-module equipped with a positive-definite bilinear form.…”

“…Van Wamelen [29] introduced an algorithm for computing Jacobi sums which is based on Stickelberger's factorization of Gauss sums [5,Thm. 11.2.2] and the LLL algorithm.…”

“…This algorithm is an enhancement and adaptation of the method described in [29,Section 7] for the fast computation of Jacobi sums. The main difference is that we are working in a subfield of the underlying cyclotomic field instead of this field itself.…”