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Algebraic Design Theory and Hadamard Matrices pp 235–249Cite as

Menon–Hadamard Difference Sets Obtained from a Local Field by Natural Projections

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 133))

Abstract

We know there exists a family of Menon–Hadamard difference sets over Galois rings of characteristic of an even power of 2 and of an odd extension degree, which has a nested structure. The projective limit of these Menon–Hadamard difference sets is a non-empty subset of a valuation ring of a local field. Conversely, does there exist a subset of a local field whose image by the natural projection always gives a difference set over a Galois ring? We will show an answer to this problem. A family of Menon–Hadamard difference sets is obtained from a subgroup of a valuation ring of a local field by the natural projections and it also has a nested structure. The formal group and the p-adic logarithm function serve an important role to the construction.

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Notes

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    This paper is in final form and no similar paper has been or is being submitted elsewhere.

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Acknowledgements

The author would like to thank the referees for the careful reading and the valuable suggestions. This work was supported by JSPS KAKENHI Grant Number 24540013.

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Authors and Affiliations

  1. School of Arts and Sciences, Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo, 167-8585, Japan

    Mieko Yamada

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  1. Mieko Yamada
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Correspondence to Mieko Yamada .

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Editors and Affiliations

  1. School of Computing, Informatics, and Decision Systems Engineering, Arizona State University, Tempe, Arizona, USA

    Charles J. Colbourn

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Dedicated to Hadi Kharaghani on the occasion on his 70th birthday

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Yamada, M. (2015). Menon–Hadamard Difference Sets Obtained from a Local Field by Natural Projections. In: Colbourn, C. (eds) Algebraic Design Theory and Hadamard Matrices. Springer Proceedings in Mathematics & Statistics, vol 133. Springer, Cham. https://doi.org/10.1007/978-3-319-17729-8_20

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