Jacobi-like sums and difference sets with Singer parameters

We show that a function F : F2m → C whose multiplicative Fourier coefficients are given by certain character sums, which we call Jacobi-like sums, is in fact the ±1-valued characteristic function of a difference set in F2m with Singer parameters. We show further that the difference sets arising from such functions are precisely the so-called DD difference sets constructed by the first author and Hans Dobbertin.

[1]  J. W. P. Hlrschfeld.  SYMMETRIC DESIGNS An Algebraic Approach (London Mathematical Society Lecture Note Series, 74) , 1984 .

[2]  M. Taylor.  INTRODUCTION TO CYCLOTOMIC FIELDS(Graduate Texts in Mathematics, 83) , 1983 .

[3]  L. D. Baumert.  Cyclic Difference Sets , 1971 .

[4]  R. Turyn.  Character sums and difference sets. , 1965 .

[5]  Jennifer D. Key,et al.  Designs and their codes , 1992, Cambridge tracts in mathematics.

[6]  Hanfried Lenz,et al.  Design theory , 1985 .

[7]  Michael Rosen,et al.  A classical introduction to modern number theory , 1982, Graduate texts in mathematics.

[8]  R. Odoni.  A CLASSICAL INTRODUCTION TO MODERN NUMBER THEORY (Graduate Texts in Mathematics, 84) , 1984 .

[9]  J. Singer.  A theorem in finite projective geometry and some applications to number theory , 1938 .

[10]  P. Kesava Menon.  Difference sets in Abelian groups , 1960 .

[11]  K. Williams,et al.  Gauss and Jacobi sums , 2021, Mathematical Surveys and Monographs.

[12]  E. Lander.  Symmetric Designs: An Algebraic Approach , 1983 .

[13]  Alexander Pott,et al.  Finite Geometry and Character Theory , 1995 .

[14]  Bernhard Schmidt,et al.  Characters and Cyclotomic Fields in Finite Geometry , 2002 .

[15]  Hans Dobbertin,et al.  New cyclic difference sets with Singer parameters , 2004, Finite Fields Their Appl..

[16]  Antonio Maschietti.  Difference Sets and Hyperovals , 1998, Des. Codes Cryptogr..

[17]  Henk D. L. Hollmann,et al.  Gauss Sums, Jacobi Sums, and p-Ranks of Cyclic Difference Sets , 1999, J. Comb. Theory, Ser. A.