Bent Partitions and Partial Difference Sets

Abstract

The recently introduced concept of a bent partition of a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2m$ </tex-math></inline-formula> -dimensional vector space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {V}_{2m}^{(p)}$ </tex-math></inline-formula> over a prime field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{p}$ </tex-math></inline-formula> exhibits similar properties as a partition from a spread. In particular, it gives rise to a large family of bent functions obtained in the same manner as spread bent functions. We show that the first non-spread construction of bent partitions introduced by Pirsic and the third author ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p=2$ </tex-math></inline-formula> ), respectively, the first and the third author ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> odd), gives rise to a large variety of different bent partitions. Especially, we show that the sets of bent functions obtained with any two such bent partitions do not intersect. We then show that every union of sets from one of these bent partitions always forms a partial difference set. This generalizes some known results on partial difference sets from spreads. Some general results on partial difference sets from bent partitions of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {V}_{2m}^{(2)}$ </tex-math></inline-formula> are given in the last section.

Topics & Concepts