A Ring Theoretic Construction of Hadamard Difference Sets in \Bbb Z 8 n\times \Bbb Z 2 n
Xiang-dong Hou
Department of Mathematics and Statistics Wright State University Dayton Ohio 45435
DOI: 10.1007/s10801-005-2512-4
Abstract
Let S= GR(2 3, n) S={\rm GR}(2^3, n) be the Galois ring of characteristic 2 3 and rank n and let R= S[ X]/( X 2, 2 X -4) R=S[X]/(X^2,\,2X-4) . We give an explicit construction of Hadamard difference sets in ( R,+) @ \Bbb Z 8 n\times \Bbb Z 2 n (R,+)\cong{\Bbb Z}_8^n\times{\Bbb Z}_2^n .}
Pages: 181–187
Keywords: keywords bent function; finite Frobenius local ring; Galois ring; Hadamard difference set
Full Text: PDF
References
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2. J.A. Davis and J. Jedwab, “A survey of Hadamard difference sets,” in Groups, Difference Sets, and the Monster, K. T. Arasu et al. (Eds.), de Gruyter, New York, 1996, pp. 145-156.
3. X. Hou, “Bent functions, partial difference sets and quasi-Frobenius local rings,” Des. Codes Cryptogr. 20 (2000), 251-268.
4. X. Hou, “Rings and constructions of partial difference sets,” Discrete Math. 270 (2003), 149-176.
5. X. Hou and A. Nechaev, A construction of finite Frobenius rings and its applications to partial difference sets, preprint.
6. R.G. Kraemer, “Proof of a conjecture on Hadamard 2-groups,” J. Combin. Theory A 63 (1993), 1-10.
7. B.R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974.
8. P.K. Menon, “On difference sets whose parameters satisfy a certain relation,” Proc. Amer. Math. Soc. 13 (1962), 739-745.
9. O.S. Rothaus, “On “bent” functions,” J. Combin. Theory A 20 (1976), 300-305.
10. R.J. Turyn, “Character sums and difference sets,” Pacific J. Math. 15 (1965), 319-346.
11. K. Yang, T. Helleseth, and P.V. Kumar, “On the weight hierarchy of Kerdock codes over Z4,” IEEE Trans. Inform. Theory 42 (1996), 1587-1593.