Multiplicative difference sets via additive characters
Recently, \textit{A. Maschietti} [Des. Codes Cryptography 14, No. 1, 89-98 (1998; Zbl 0887.50010)] proposed a new construction of difference sets in the finite field \(\mathbb{F}_{2^m}\). It works by choosing a positive integer \(k\) satisfying two conditions, (1) that \(x\mapsto x^k\) is a permutation on \(\mathbb{F}_{2^m}\), and (2) that \(x\mapsto x+x^k\) is a 2-to-1 mapping on \(\mathbb{F}_{2^m}\). Then \(\{x+x^k:x\in \mathbb{F}_{2^m}\}\setminus \{0\}\) is a difference set in the multiplicative group \(\mathbb{F}_{2^m}^\times\) of \(\mathbb{F}_{2^m}\). \textit{J.-S. No, H. Chung} and \textit{M.-S. Yun} [IEEE Trans. Inf. Theory 44, No. 3, 1278-1282 (1998; Zbl 0912.94017)] considered a variation of that construction: They looked at sets arising from maps of the form \(x\mapsto x^d+(x+1)^d\) and conjectured that if \(m=3k\pm 1\) and \(d=4^k-2^k+1\), then \(\{x^d+(x+1)^d:x\in \mathbb{F}_{2^m}\}\setminus \{0\}\) is a difference set in \(\mathbb{F}_{2^m}^\times\). In the paper under review, not only a new proof of Maschietti's result is given, based on computations with the autocorrelation function and the use of Fourier analysis in the additive group of \(\mathbb{F}_{2^m}\), but also, using similar ideas, the conjecture by No, Chung and Yun is proved for \(m\) odd. Along the way, a stronger form of a celebrated theorem of Welch on the 3-valued cross-correlation of maximal length sequences is proved.
scientific article; zbMATH DE number 1392714