Designs, Codes and Cryptography

, Volume 86, Issue 12, pp 2857–2868

# Some results on generalized strong external difference families

• Xiaojuan Lu
• Xiaolei Niu
• Haitao Cao
Article

## Abstract

A generalized strong external difference family (briefly $$(v, m; k_1,\dots ,k_m; \lambda _1,\dots ,\lambda _m)$$-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we give some nonexistence results for GSEDFs. In particular, we prove that a $$(v, 3;k_1,k_2,k_3; \lambda _1,\lambda _2,\lambda _3)$$-GSEDF does not exist when $$k_1+k_2+k_3< v$$. We also give a first recursive construction for GSEDFs and prove that if there is a $$(v,2;2\lambda ,\frac{v-1}{2};\lambda ,\lambda )$$-GSEDF, then there is a $$(vt,2;4\lambda ,\frac{vt-1}{2};2\lambda ,2\lambda )$$-GSEDF with $$v>1$$, $$t>1$$ and $$v\equiv t\equiv 1\pmod 2$$. Then we use it to obtain some new GSEDFs for $$m=2$$. In particular, for any prime power q with $$q\equiv 1\pmod 4$$, we show that there exists a $$(qt, 2;(q-1)2^{n-1},\frac{qt-1}{2};(q-1)2^{n-2},(q-1)2^{n-2})$$-GSEDF, where $$t=p_1p_2\dots p_n$$, $$p_i>1$$, $$1\le i\le n$$, $$p_1, p_2,\dots ,p_n$$ are odd integers.

## Keywords

Generalized strong external difference family Difference set Character theory Nonexistence

05B05 05B10

## Notes

### Acknowledgements

We would like to thank Prof. K. Feng of Tsinghua university for suggesting this research topic. We also thank the anonymous referees for their careful reading of the manuscript and many constructive comments and suggestions that greatly improved the readability of this paper.

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## Authors and Affiliations

• Xiaojuan Lu
• 1
• Xiaolei Niu
• 1
• Haitao Cao
• 1
1. 1.Institute of MathematicsNanjing Normal UniversityNanjingChina