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Divisible difference families from Galois rings \(GR(4,n)\) and Hadamard matrices

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Abstract

We give a new construction of difference families generalizing Szekeres’s difference families Szekeres (Enseignment Math 15:269–278, 1969). As an immediate consequence, we obtain some new examples of difference families with several blocks in multiplicative subgroups of finite fields. We also prove that there exists an infinite family of divisible difference families with two blocks in a unit subgroup of the Galois ring \(GR(4,n)\). Furthermore, we obtain a new construction method of symmetric Hadamard matrices by using divisible difference families and a new array.

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Acknowledgments

The work of K. Momihara was supported by JSPS KAKENHI Grant Number 23840032 and the work of M. Yamada was supported by JSPS KAKENHI Grant Number 90540014. The authors thank the referees for helpful comments and suggestions that helped to improve the presentation of the paper.

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Authors and Affiliations

  1. Faculty of Education, Kumamoto University, 2-40-1 Kurokami, Kumamoto, 860-8555, Japan

    Koji Momihara

  2. Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, 920-1192, Japan

    Mieko Yamada

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  1. Koji Momihara
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  2. Mieko Yamada
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Correspondence to Koji Momihara.

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Communicated by K. T. Arasu.

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Momihara, K., Yamada, M. Divisible difference families from Galois rings \(GR(4,n)\) and Hadamard matrices. Des. Codes Cryptogr. 73, 897–909 (2014). https://doi.org/10.1007/s10623-013-9833-4

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  • DOI: https://doi.org/10.1007/s10623-013-9833-4

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