Browsing by Author "Xiang, Qing"
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Item Cyclic Relative Difference Sets and Their p-Ranks(Department of Mathematical Sciences, 2002) Chandler, D.B.; Xiang, QingBy modifying the constructions in [10] and [15], we construct a family of cyclic ((q 3k − 1)/(q − 1), q − 1, q 3k − 1 , q 3k − 2 ) relative difference sets, where q = 3 e . These relative difference sets are “liftings” of the difference sets constructed in [10] and [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q = 3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.Item The Invariant Factors of Some Cyclic Difference Sets(Department of Mathematical Sciences, 2002) Chandler, D.B.; Xiang, QingUsing the Smith normal forms of the symmetric designs associated with the HKM and Lin difference sets, we show that not only are these two families of difference sets inequivalent, but also that the associated symmetric designs are nonisomorphic.Item On Mathon's Construction of Maximal Arcs in Desarguesian Planes(Department of Mathematical Sciences, 2002) Fiedler, F.; Leung, K.H.; Xiang, QingWe study the problem of determining the largest d of a non-Denniston max-imal arc of degree 2 d generated by a { p, 1 } -map in PG(2, 2 m ) via a recent construction of Mathon [M]. On one hand, we show that there are { p, 1 } -maps that generate non-Denniston maximal arcs of degree 2 m+1 2 , where m 5 is odd. Together with Mathon’s result [M] in the m even case, this shows that there are always { p, 1 } -maps generating non-Denniston maximal arcs of degree 2 b m+2 2 c in PG(2, 2 m ). On the other hand, we prove that the largest degree of a non-Denniston maximal arc in PG(2, 2 m ) constructed using a { p, 1 } -map is less than or equal to 2 m − 3 . We conjecture that this largest degree is actually 2 b m+2 2 c .