Cyclic Relative Difference Sets with Classical Parameters

Constructions of Relative Difference Sets with Classical Parameters and Circulant Weighing Matrices

“…In fact, a new construction of cyclic relative difference sets with classical parameters was found where q is a 2-power, d is odd and n ¼ 2ðq À 1Þ: This helps us to answer Pott's question when the group is cyclic. Theorem 1.2 (Arasu et al [2]). Let q be a prime power.…”

“…In this paper, we investigate further 'lifting' of the relative difference sets constructed in [2]. We show that under some conditions, n can go up to 4ðq À 1Þ: More specifically, we construct a family of relative difference sets with classical parameters when q is a 2-power, d ¼ 7 and n ¼ 4ðq À 1Þ: Obviously, in view of Theorem 1.2, the groups where the new relative difference sets lie in are no longer cyclic.…”

“…We show that under some conditions, n can go up to 4ðq À 1Þ: More specifically, we construct a family of relative difference sets with classical parameters when q is a 2-power, d ¼ 7 and n ¼ 4ðq À 1Þ: Obviously, in view of Theorem 1.2, the groups where the new relative difference sets lie in are no longer cyclic. In fact, the group concerned is the direct product of an even order cyclic group and a copy of Z 2 : The main idea of our construction is to combine several relative difference sets constructed in [2].…”

“…48] asked if there exists a relative ðm; n; k; lÞ-difference set such that its projection is the complement of a Singer difference set, n=2 and n is not a divisor of q À 1: In [2], the cyclic case was studied. In fact, a new construction of cyclic relative difference sets with classical parameters was found where q is a 2-power, d is odd and n ¼ 2ðq À 1Þ: This helps us to answer Pott's question when the group is cyclic.…”

“…The construction is based on the technique used in [2]. By a similar method, we also construct some new circulant weighing matrices of order q dÀ1 where q is a 2-power, d is odd and d55: # 2002 Elsevier Science (USA) …”