On ( p a, p b, p a, p a-b)-Relative Difference Sets
Bernhard Schmidt
DOI: 10.1023/A:1008674331764
Abstract
This paper provides new exponent and rank conditions for the existence of abelian relative ( p a, p b, p a, p a-b)-difference sets. It is also shown that no splitting relative (2 2c,2 d,2 2c,2 2c-d)-difference set exists if d > c and the forbidden subgroup is abelian. Furthermore, abelian relative (16, 4, 16, 4)-difference sets are studied in detail; in particular, it is shown that a relative (16, 4, 16, 4)-difference set in an abelian group G 
Z 8 \times Z 4 \times Z 2 exists if and only if exp( G) 
4 or G = Z 8 \times (Z 2) 3 with N 
Z 2 \times Z 2.
Z 8 \times Z 4 \times Z 2 exists if and only if exp( G) 4 or G = Z 8 \times (Z 2) 3 with N Z 2 \times Z 2.Pages: 279–297
Keywords: relative difference set; exponent bounds; abelian character
Full Text: PDF
References
1. T. Beth, D. Jungnickel, and H. Lenz, Design Theory, Cambridge University Press, Cambridge, 1986.
2. R.C. Bose, “An affine analogue of Singer's theorem,” J. Indian Math. Soc. 6 (1942), 1-15.
3. B.W. Brock, “A new construction of circulant GH( p2, Zp),” Discrete Math. 112 (1993), 249-252.
4. J.A. Davis, “A note on products of relative difference sets,” Designs, Codes and Cryptography 1 (1991), 117-119.
5. J.A. Davis, “Construction of relative difference sets in p-groups,” Discrete Math. 103 (1992), 7-15.
6. J.A. Davis, “An exponent bound for relative difference sets in p-groups,” Ars. Comb. 34 (1992), 318-320.
7. J.A. Davis and S.K. Seghal, “Using the simplex code to construct relative difference sets in 2-groups,” Designs, Codes and Cryptography (1994), submitted.
8. W. de Launey and P. Viyaj Kumar, “On circulant generalized Hadamard matrices of prime power order,” Designs, Codes and Cryptography (1994), (to appear).
9. J.E.H. Elliott and A.T. Butson, “Relative difference sets,” Illinois J. Math. 10 (1966), 517-531.
10. M.J. Ganley, “On a paper of Dembowski and Ostrom,” Arch. Math. 27 (1976), 93-98.
11. A.J. Hoffman, “Cyclic affine planes,” Can. J. Math. 4 (1952), 135-145.
12. D. Jungnickel, “On a theorem of Ganley,” Graphs and Comb. 3 (1987), 141-143.
13. K.H. Leung and S.L. Ma, “Constructions of partial difference sets and relative difference sets on p-groups,” Bull. Lond. Math. Soc. (1990), 533-539. P1: MBL/SNG P2: ICA/SFI P3: RPS/SFI QC: ICA Journal of Algebraic Combinatorics KL434-07-Schmidt April 23, 1997 16:19 ON ( pa, pb, pa, pa - b)-RELATIVE DIFFERENCE SETS 297
14. S.L. Ma, Polynomial Addition Sets, Ph.D. Thesis, University of Hong Kong, 1985.
15. S.L. Ma and A. Pott, “Relative difference sets, planar functions and generalized Hadamard matrices,” J. Algebra 175 (1995), 505-525.
16. S.L. Ma and B. Schmidt, “On ( pa, p, pa, pa - 1)-relative difference sets,” Designs, Codes and Cryptography 6 (1995), 57-72.
17. S.L. Ma and B. Schmidt, “The structure of abelian groups containing McFarland difference sets,” J. Comb. Theory A70 (1995), 313-322.
18. A. Pott, “On the structure of abelian groups admitting divisible difference sets,” J. Combin. Theory A2 (1994), 202-213.
19. A. Pott, “A survey on relative difference sets,” in Groups, Difference Sets and the Monster, K.T. Arasu, J.F. Dillon, K. Harada, S.K. Seghal, and R.L. Solomon (Eds.), DeGruyter Verlag Berlin/New York, pp. 195-233, 1996.
20. B. Schmidt, “Abelian (16,4,16,4)-RDS,” manuscript.
21. R.J. Turyn, “Character sums and difference sets,” Pacific J. Math. 15 (1965), 319-346.
2. R.C. Bose, “An affine analogue of Singer's theorem,” J. Indian Math. Soc. 6 (1942), 1-15.
3. B.W. Brock, “A new construction of circulant GH( p2, Zp),” Discrete Math. 112 (1993), 249-252.
4. J.A. Davis, “A note on products of relative difference sets,” Designs, Codes and Cryptography 1 (1991), 117-119.
5. J.A. Davis, “Construction of relative difference sets in p-groups,” Discrete Math. 103 (1992), 7-15.
6. J.A. Davis, “An exponent bound for relative difference sets in p-groups,” Ars. Comb. 34 (1992), 318-320.
7. J.A. Davis and S.K. Seghal, “Using the simplex code to construct relative difference sets in 2-groups,” Designs, Codes and Cryptography (1994), submitted.
8. W. de Launey and P. Viyaj Kumar, “On circulant generalized Hadamard matrices of prime power order,” Designs, Codes and Cryptography (1994), (to appear).
9. J.E.H. Elliott and A.T. Butson, “Relative difference sets,” Illinois J. Math. 10 (1966), 517-531.
10. M.J. Ganley, “On a paper of Dembowski and Ostrom,” Arch. Math. 27 (1976), 93-98.
11. A.J. Hoffman, “Cyclic affine planes,” Can. J. Math. 4 (1952), 135-145.
12. D. Jungnickel, “On a theorem of Ganley,” Graphs and Comb. 3 (1987), 141-143.
13. K.H. Leung and S.L. Ma, “Constructions of partial difference sets and relative difference sets on p-groups,” Bull. Lond. Math. Soc. (1990), 533-539. P1: MBL/SNG P2: ICA/SFI P3: RPS/SFI QC: ICA Journal of Algebraic Combinatorics KL434-07-Schmidt April 23, 1997 16:19 ON ( pa, pb, pa, pa - b)-RELATIVE DIFFERENCE SETS 297
14. S.L. Ma, Polynomial Addition Sets, Ph.D. Thesis, University of Hong Kong, 1985.
15. S.L. Ma and A. Pott, “Relative difference sets, planar functions and generalized Hadamard matrices,” J. Algebra 175 (1995), 505-525.
16. S.L. Ma and B. Schmidt, “On ( pa, p, pa, pa - 1)-relative difference sets,” Designs, Codes and Cryptography 6 (1995), 57-72.
17. S.L. Ma and B. Schmidt, “The structure of abelian groups containing McFarland difference sets,” J. Comb. Theory A70 (1995), 313-322.
18. A. Pott, “On the structure of abelian groups admitting divisible difference sets,” J. Combin. Theory A2 (1994), 202-213.
19. A. Pott, “A survey on relative difference sets,” in Groups, Difference Sets and the Monster, K.T. Arasu, J.F. Dillon, K. Harada, S.K. Seghal, and R.L. Solomon (Eds.), DeGruyter Verlag Berlin/New York, pp. 195-233, 1996.
20. B. Schmidt, “Abelian (16,4,16,4)-RDS,” manuscript.
21. R.J. Turyn, “Character sums and difference sets,” Pacific J. Math. 15 (1965), 319-346.