On the Symmetric Difference Property in Difference Sets under Product Construction

TL;DR

This paper investigates how the symmetric difference property (SDP) in symmetric designs is preserved under product constructions of difference sets, with implications for coding theory.

Contribution

It proves that the SDP is maintained when difference sets are combined via direct product construction and explores isomorphism properties of the resulting designs.

Findings

SDP is preserved under direct product construction of difference sets

Product constructed SDP designs retain the symmetric difference property

Results on isomorphisms in product SDP designs are established

Abstract

A $(v, k, \lambda)$ symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. Symmetric designs fulfilling this property have the nice property of having minimal rank, which makes them interesting to study. Thus, SDP designs become useful in coding theory applications. We show in this paper that difference sets formed by direct product construction of difference sets whose developments have the SDP also have the SDP. We also establish a few results regarding isomorphisms in product constructed SDP designs.