Premium
Skew Hadamard Difference Sets from Dickson Polynomials of Order 7
Author(s) -
Ding Cunsheng,
Pott Alexander,
Wang Qi
Publication year - 2015
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21421
Subject(s) - mathematics , hadamard transform , skew , combinatorics , difference set , hadamard three lines theorem , order (exchange) , conjecture , hadamard matrix , hadamard's inequality , hadamard three circle theorem , hadamard's maximal determinant problem , abelian group , discrete mathematics , mathematical analysis , physics , finance , astronomy , economics
Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in F q where q ≡ 3 mod 4 ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set ofD 5 ( x 2 , u )is a new skew Hadamard difference set in ( F 3 m , + ) with m odd, whereD n ( x , u )denotes the first kind of Dickson polynomials of order n and u ∈ F q * . The key observation in the proof is thatD 5 ( x 2 , u )is a planar function from F 3 mto F 3 mfor m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all u ∈ F 3 m * , the setD u : = { D 7 ( x 2 , u ) : x ∈ F 3 m * }is a skew Hadamard difference set in ( F 3 m , + ) , where m is odd andm ¬ ≡ 0( mod 3 ) . The proof is more complicated and different than that of Ding‐Yuan skew Hadamard difference sets sinceD 7 ( x 2 , u )is not planar in F 3 m . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for m = 5 , 7 by comparing the triple intersection numbers.