Difference Sets with n = 2pm
Mikhail Muzychuk
DOI: 10.1023/A:1008671228817
Abstract
Let D be a (v,k, 
) difference set over an abelian group G with even n = k - . Assume that t 
N satisfies the congruences t 
q i fi (mod exp(G)) for each prime divisor q i of n/2 and some integer f i. In [4] it was shown that t is a multiplier of D provided that n > , (n/2, ) = 1 and (n/2, v) = 1. In this paper we show that the condition n > may be removed. As a corollary we obtain that in the case of n= 2p a when p is a prime, p should be a multiplier of D. This answers an open question mentioned in [2].
) difference set over an abelian group G with even n = k - . Assume that t N satisfies the congruences t q i fi (mod exp(G)) for each prime divisor q i of n/2 and some integer f i. In [4] it was shown that t is a multiplier of D provided that n > , (n/2, ) = 1 and (n/2, v) = 1. In this paper we show that the condition n > may be removed. As a corollary we obtain that in the case of n= 2p a when p is a prime, p should be a multiplier of D. This answers an open question mentioned in [2].Pages: 77–89
Keywords: difference set; abelian group
Full Text: PDF
References
1. C.W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, John Wiley & Sons, New York, London, 1962.
2. D. Jungnickel, “Difference Sets,” in Contemporary Design Theory: A Collection of Surveys, J.H. Dinitz and D.R. Stinson (Eds.), John Wiley & Sons, pp. 241-324, 1992.
3. H.B. Mann, Addition Theorems, Wiley, New York, 1965.
4. H.B. Mann and S.K. Zaremba, “On multipliers of difference sets,” Illinois J. Math. 13 (1969), 378-382.
5. Qiu Weisheng, “On character approach to multiplier conjecture and a new result on it,” 1993, submitted.
2. D. Jungnickel, “Difference Sets,” in Contemporary Design Theory: A Collection of Surveys, J.H. Dinitz and D.R. Stinson (Eds.), John Wiley & Sons, pp. 241-324, 1992.
3. H.B. Mann, Addition Theorems, Wiley, New York, 1965.
4. H.B. Mann and S.K. Zaremba, “On multipliers of difference sets,” Illinois J. Math. 13 (1969), 378-382.
5. Qiu Weisheng, “On character approach to multiplier conjecture and a new result on it,” 1993, submitted.