Difference sets and tri-weight linear codes from trinomials over binary fields

TL;DR

This paper proves that certain trinomials over binary fields produce difference sets with specific parameters and that associated linear codes are tri-weight, advancing understanding in combinatorial design and coding theory.

Contribution

It confirms conjectures linking trinomials over binary fields to difference sets and tri-weight codes, revealing new properties of their value-sets and code structures.

Findings

Value-sets have elements with one or four preimages.

The eleven trinomials produce difference sets with Singer parameters.

Constructed codes are confirmed to be tri-weight.

Abstract

We confirm a conjecture of Cun Sheng Ding~\cite{Ding-Discrete} 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 preimages. We also give a partial resolution of another conjecture of Cun Sheng Ding~\cite{Ding-Discrete} claiming that linear codes constructed from those eleven trinomials are tri-weight.