Cyclotomy and addition sets (Q1238814)

\(A(v,k,\lambda,g)\) - addition set \(A=(a_1,\dots,a_k)\) is a collection of \(k\) distinct residues modulo \(v\) such that any non-zero residue \(\gamma\) has exactly \(\lambda\) representations of the form \(a_i+ga_j\equiv\gamma\). If the polynomial \(\theta(x)=x^{a_1}+\dots+x^{a_k}\) is considered this leads to \(\theta(x)\theta(x^g)\equiv d+\lambda(1+x+\dots+x^{v-1})\) modulo \(x^v-1\), where \(d+\lambda\) is the number of representations of \(0\). Motivated by the application to addition sets the author studies the existence question for such polynomials \(\theta(x)\), which naturally leads to considerations of multipliers and to cyclotomy. He searches for addition sets comprised of unions of index classes for primes \(p=Nf+1\) with \(N\leq12\) and finds some for \(N=6,8,10\) and \(12\). In fact, for \(N=8\) an infinite class is constructed and for \(N=10\), \(p=41\), he finds an example with \(1<d\neq k-\lambda\) which he claims is the first such known. Using the machinery developed earlier he then examines two cases remaining from his earlier studies, (95,10,1,18) and (95,10,1,56); he shows that no such addition sets exist.