In this paper, we construct an infinite family of hemisystems of the Hermitian surface H(3, q 2 ). In particular, we show that for every odd prime power q congruent to 3 modulo 4, there exists a hemisystem of H(3, q 2 ) admitting C (q 3 +1)/4 : C 3 .2010 Mathematics Subject Classification. 05B25 (primary), 05E30, 51E12 (secondary).
In this paper, we give an algebraic construction of a new infinite family of Cameron-Liebler line classes with parameter x = q 2 −1 2 for q ≡ 5 or 9 (mod 12), which generalizes the examples found by Rodgers in [26] through a computer search. Furthermore, in the case where q is an even power of 3, we construct the first infinite family of affine two-intersection sets in AG(2, q), which is closely related to our Cameron-Liebler line classes.
In this paper, we give a construction of strongly regular Cayley graphs and a construction of skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields, and they generalize the constructions given by Feng and Xiang [10,12]. Three infinite families of strongly regular graphs with new parameters are obtained. The main tools that we employed are index 2 Gauss sums, instead of cyclotomic numbers.
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