Difference sets and three-weight linear codes from trinomials

Abstract

We confirm a conjecture of Cunsheng Ding claiming that the punctured value-sets of a list of eleven trinomials over odd-degree extensions of the binary field give rise to difference sets with Singer parameters. In the course of confirming the conjecture, we show that these trinomials share the remarkable property that every element of the value-set of each trinomial has either one or four preiamges. We also give the partial resolution of another conjecture of Cunsheng Ding claiming that linear codes constructed from those eleven trinomials are three-weight.