Cyclotomy and addition sets

Abstract

In this paper, we develop methods to solve the polynomial congruence θ( x) θ( x g ) ≡ d + λ(1 + x +… + x p−1 ) (mod x p − 1), where p is an odd prime and θ(x) is a polynomial with nonnegative integral coefficients. Using these methods, we construct some new addition sets that are the unions of index classes for some primes p. We also establish the nonexistence of both the (95, 10, 1, 18)-addition set and the (95, 10, 1, 56)-addition set.