Gauss Sums, Jacobi Sums, and p-Ranks of Cyclic Difference Sets

Journal of Combinatorial Theory. Series A

Volumn 87, Issue 1, 1999, Pages 74-119

  Evans, Ronald ^a   Hollmann, Henk D L ^b   Krattenthaler, Christian ^c   Xiang, Qing ^d  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b PHILIPS RESEARCH LABORATORIES   (Netherlands)

^c UNIVERSITY OF VIENNA   (Austria)

^d UNIVERSITY OF DELAWARE   (United States)

Author keywords

Binary cyclic code; Cyclic difference set; Finite fields; Gauss sum; Glynn hyperoval; GMW difference set; Jacobi sum; Monomial hyperoval; Quadratic residue difference set; Segre hyperoval; Singer difference set; Teichm ller character

Indexed keywords

EID: 0002248212     PISSN: 00973165     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcta.1998.2950     Document Type: Article

References (33)

1. m) with a two-level autocorrelation function and a large linear span, IEEE Trans. Inform. Theory 38 (1992), 120-130.
2. E. F. Assmus, Jr., and J. D. Key, "Designs and Their Codes," Cambridge Tracts in Mathematics, Vol. 103, Cambridge Univ. Press, Cambridge, UK, 1992.
3. B. C. Berndt and R. J. Evans, Sums of Gauss, Jacobi, and Jacobsthal, J. Number Theory 11 (1979), 349-398.
4. B. C. Berndt, R. J. Evans, and K. S. Williams, "Gauss and Jacobi Sums," Wiley-Interscience, New York, 1998.
5. W. G. Bridges, M. Hall, and J. L. Hayden, Codes and designs, J. Combin. Theory Ser. A 31 (1981), 155-174.
6. J. Cannon and C. Playoust, "An Introduction to MAGMA," University of Sydney, Australia, 1993.
7. W. E. Cherowitzo and L. Storme, α-flocks with oval herds and monomial hyperovals, Finite Fields Appl. 4 (1998), 185-199.
8. D. Glynn, Two new sequences of ovals in finite Desarguesian planes of even order, in "Lecture Notes in Mathematics," Vol. 1036, pp. 217-229, Springer-Verlag, New York/ Berlin, 1983.
9. J. M. Goethals and P. Delsarte, On a class of majority logic decodable cyclic codes, IEEE Trans. Inform. Theory 14 (1968), 182-188.
10. S. W. Golomb, The use of combinatorial structures in communication signal designs, in "Applications of Combinatorial Mathematics" (Chris Mitchell, Ed.), Vol. 60, pp. 59-78, 1997.
11. B. Gordon, W. H. Mills, and L. R. Welch, Some new difference sets, Canad. J. Math. 14 (1962), 614-625.
12. N. Hamada, On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error-correcting codes, Hiroshima Math. J. 3 (1973), 154-226.
13. N. Hamada and H. Ohmori, On the BIB-design having the minimum p-rank, J. Combin. Theory Ser. A 18 (1975), 131-140.
14. J. W. P. Hirschfeld, "Projective Geometries over Finite Fields," Oxford Univ. Press, London, 1979.
15. D. Jungnickel, Difference sets, in "Contemporary Design Theory, a Collection of Surveys" (J. Dinitz and D. R. Stinson, Eds.), Wiley-Interscience Series in Discrete Mathematics and Optimization, pp. 241-324, Wiley, New York, 1992.
16. E. S. Lander, "Symmetric Designs, an Algebraic Approach," Cambridge Univ. Press, Cambridge, UK, 1983.
17. R. Lidl and H. Niederreiter, "Finite Fields," 2nd ed., Encyclopedia of Mathematics and Its Applications, Vol. 20, Cambridge Univ. Press, Cambridge, UK, 1997.
18. M. Lothaire, "Combinatorics on Words," 2nd ed., Encyclopedia of Mathematics and Its Applications, Vol. 17, Cambridge Univ. Press, Cambridge, UK, 1997.
19. J. MacWilliams and H. B. Mann, On the p-rank of the design matrix of a difference set, Inform. Control 12 (1968), 474-488.
20. F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Encoding Codes," North-Holland, Amsterdam, 1977.
21. A. Maschietti, Difference sets and hyperovals, Des. Codes Cryptogr. 14 (1998), 89-98.
22. S. E. Payne and J. A. Thas, "Finite Generalized Quadrangles," Research Notes in Mathematics, Vol. 110, Pitman, London, 1984.
23. B. Salvy and P. Zimmermann, Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (1994).
24. R. A. Scholtz and L. R. Welch, GMW sequences, IEEE Trans. Inform. Theory 30 (1984), 548-553.
25. B. Segre, Ovals in a finite projective plane, Canad. J. Math. 7 (1955), 414-416.
26. K. J. C. Smith, On the p-rank of the incidence matrix of points and hyperplanes in a finite projective geometry, J. Combin. Theory 7 (1969), 122-129.
27. R. P. Stanley, "Enumerative Combinatorics," Vol. 1, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1986; reprint, Cambridge Univ. Press, Cambridge, UK, 1997.
28. R. J. Turyn, Character sums and difference sets, Pacific J. Math. 15 (1965), 319-346.
29. L. C. Washington, "Introduction to Cyclotomic Fields," 2nd ed., Springer-Verlag, New York/Berlin, 1997.
30. H. S. Wilf, "Generatingfunctionology," 2nd ed., Academic Press, San Diego, 1994.
31. Q. Xiang, On balanced binary sequences with two-level autocorrelation functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.
32. K. Yamamoto, On Jacobi sums and difference sets, J. Combin. Theory Ser. A 3 (1967), 146-181.
33. K. Yamamoto, On congruences arising from relative Gauss sums, in "Number Theory and Combinatorics," pp. 423-446, World Scientific, Singapore, 1955.