#### DOI

10.1007/BF01299738

#### Abstract

A Menon difference set has the parameters (4*N*^{2}, *2N*^{2}-*N, N*^{2}-*N*). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group H×K×Z_{pα} contains a Menon difference set, where *p* is an odd prime, |K|=*p*^{α}, and *p*^{j}≡−1 (mod exp (*H*)) for some *j*. Using the viewpoint of perfect binary arrays we prove that *K* must be cyclic. A corollary is that there exists a Menon difference set in the abelian group H×K×Z_{3α}, where exp (*H*)=2 or 4 and |K|=3^{α}, if and only if *K* is cyclic.

#### Document Type

Article

#### Publication Date

9-1995

#### Publisher Statement

Copyright © 1995, Springer-Verlag. This article first appeared in *Combinatorica*: 15:3 (1995), 311-317.

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#### Recommended Citation

Arasu, K. T., James A. Davis, and Jonathan Jedwab. "A Nonexistence Result for Abelian Menon Difference Sets Using Perfect Binary Arrays." *Combinatorica* 15, no. 3 (September 1995): 311-17. doi:10.1007/BF01299738.