On the existence of (400, 57, 8) non-abelian difference sets

Difference sets with parameters (\frac{qd + 1 - 1}{q - 1}, \frac{qd - 1}{q - 1}, \frac{qd - 1 - 1}{q - 1}), where q is a prime power and d \geq 1, are known to exist in cyclic groups and are called classical Singer difference sets. We study a special case of this family with q = 7 and d = 3 in search of more difference sets. According to GAP, there are 220 groups of order 400 out of which 10 are abelian. E. Kopilovich and other authors showed that the remaining nine abelian groups of order 400 do not admit (400, 57, 8) difference sets. Also, Gao and Wei used the (400, 57, 8) Singer difference set to construct four inequivalent difference sets in a non-abelian group. In this paper, we demonstrate using group representation and factorization in cyclotomic rings that, out of the remaining 209 non-abelian groups of order 400, only 15 could possibly admit (400, 57, 8) difference sets.

Anahtar Kelimeler:

Representation, idempotents, Singer difference sets, intersection numbers

On the existence of (400, 57, 8) non-abelian difference sets

Keywords:

Representation, idempotents, Singer difference sets, intersection numbers,

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Baumert, L. D.: Cyclic Diﬀerence Sets, Springer-Verlag Publishers (1971).
Dillon, J.: Variations on a scheme of McFarland for noncyclic diﬀerence sets. J. Comb. Theory A 40, 9–21 (1985). Gao, S., and Wei, W.: On non-Abelian group diﬀerence sets. Discrete Math. 112, 93–102 (1993).
GAP Group: Groups, Algorithms and Programming, Version 4. 4. 6. Retrieved on July 30 2009 from: http://www.gap-system.org/Download/index.html.
Gjoneski, O., Osifodunrin, A. S., and Smith, K. W.: Non existence of (176, 50, 14) and (704, 38, 2) diﬀerence sets, to appear. Iiams, J. E.: Lander’s Tables are complete. Diﬀerence sets, sequences and their correlation properties. Kluwer, 239– 257 (1999).
Kopilovich, L. E.: Diﬀerence sets in non cyclic abelian groups. Cybernetics 25, 153–157 (1996).
Lander, E.: Symmetric design: an algebraic approach, London Math. Soc. Lecture Note Series 74, Cambridge Univ. Press 1983.
Lang, S.: Algebraic number theory, Reading, MA-USA, Addison-Wesley 1970.
Ledermann, W.: Introduction to Group Characters, Cambridge Univ. Press 1977.
Leung, K. H., Ma, S. L., and Schmidt, B.: Non-existence of abelian sets: Lander’s conjecture for prime power orders. American Mathematical Society 356(11), 43–58 (2003).
Liebler, R.: The Inversion Formula. J. Combin. Math. and Combin. Computing 13, 143–160 (1993).
Ma, S. L.: Planar functions, relative diﬀerence sets and character theory. J. of Algebra 185, 342–356 (1996).
Osifodunrin, A. S.: Investigation of Diﬀerence Sets With Order 36, PhD dissertation, Central Michigan University, Mount Pleasant, MI, May, 2008.
Osifodunrin, A. S., and Smith, K. W.: On the existence of (160, 54, 18) diﬀerence sets, to appear. Pott, A.: Finite Geometry and Character Theory, Springer-Verlag Publishers 1995.
Schmidt, B.: Cyclotomic Integers and Finite Geometry. Jour. of Ame. Math. Soc. 12(4), 929–952 (1999).
Smith, K. W.: Non-Abelian Hadamard Diﬀerence sets. J. Comb. Theory A, 144–156 (1995).
Turyn, R.: Character sums and diﬀerence set, Paciﬁc J. Math. 15, 319–346 (1965).