Difference Balanced Functions and Their Generalized Difference Sets

Difference balanced functions from $F_{q^n}^*$ to $F_q$ are closely related to combinatorial designs and naturally define $p$-ary sequences with the ideal two-level autocorrelation. In the literature, all existing such functions are associated with the $d$-homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be $d$-homogeneous. First we characterize difference balanced functions by generalized difference sets with respect to two exceptional subgroups. We then derive several necessary and sufficient conditions for $d$-homogeneous difference balanced functions. In particular, we reveal an unexpected equivalence between the $d$-homogeneous property and multipliers of generalized difference sets. By determining these multipliers, we prove the Gong-Song conjecture for $q$ prime. Furthermore, we show that every difference balanced function must be balanced or an affine shift of a balanced function.Comment: 17 page

The design research pyramid: a three layer framework

To support knowledge-based design development, considerable research has been conducted from various perspectives at different levels. The research on knowledge-based design support systems, generic design artefact and design process modelling, and the inherent quality of design knowledge itself are some examples of these perspectives. The structure underneath the research is not a disparate one but ordered. This paper provides an overview of some ontologies of design knowledge and a layered research framework of knowledge-based engineering design support. Three layers of research are clarified in this pattern: knowledge ontology, design knowledge model, and application. Specifically, the paper highlights ontologies of design knowledge by giving a set of classifications of design knowledge from different points of view. Within the discussion of design knowledge content ontology, two topologies, i.e., teleology and evolutionary, are identified