Two Generalized Constructions of Relative Difference Sets
Xiang-Dong Hou
and Surinder K. Sehgal
DOI: 10.1023/A:1026540010181
Abstract
We give two generalizations of some known constructions of relative difference sets. The first one is a generalization of a construction of RDS by Chen, Ray-Chaudhuri and Xiang using the Galois ring GR(4, m). The second one generalizes a construction of RDS by Ma and Schmidt from the setting of chain rings to a setting of more general rings.
Pages: 145–153
Keywords: bent function; exponential sum; finite quasi-Frobenius local ring; relative difference set
Full Text: PDF
References
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2. Y. Chen, D.K. Ray-Chaudhuri, and Q. Xiang, “Constructions of partial difference sets and relative difference sets using Galois rings II,” J. Combin. Theory, Ser A 76 (1996), 179-196
3. J.A. Davis and J. Jedwab, “A unifying construction for difference sets,” J. Combin. Theory, Ser A 80 (1997), 13-78.
4. L.E. Dickson, Linear Groups with an Exposition of the Galois Field Theory, Dover, New York, 1958.
5. A.R. Hammons, Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Solé, “The Z4-linearity of Kerdock, Preparata, Goethals, and related codes,” IEEE Trans. Inform. Theory 40 (1994), 301-319.
6. X. Hou, “Bent functions, partial difference sets, and quasi-Frobenius local rings,” Designs, Codes and Cryptgor., 20 (2000), 251-268.
7. X. Hou and S.K. Sehgal, “An extension of building sets,” J. Combin. Designs 8 (2000), 50-57.
8. S.L. Ma and B. Schmidt, “Relative ( pa, pb, pa, pa - b)-difference sets: A unified exponent bound and a local ring construction,” preprint.
9. B.R. McDonald, Finite Rings with Identity, Dekker, New York, (1974).
10. A. Pott, “A survey on relative difference sets,” in Groups, Difference Sets, and the Monster, K.T. Arasu et al. (eds.), de Gruyter, Berlin, 1996, pp. 195-232.
11. O.S. Rothaus, “On “bent” functions,” J. Combin. Theory, Ser A 20 (1976), 300-305.
12. K. Yang, T. Helleseth, P.V. Kumar, and A.G. Shanbhag, “On the weight hierarchy of Kerdock codes over Z4,” IEEE Trans. Inform. Theory 42 (1996), 1587-1593.