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Designs, Codes and Cryptography

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28-01-2022

Difference matrices with five rows over finite abelian groups

Authors: Rong Pan, R. Julian R. Abel, Yudhistira A. Bunjamin, Tao Feng, Tiana J. Tsang Ung, Xiaomiao Wang

Published in: Designs, Codes and Cryptography | Issue 2/2022

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Abstract

Let G be a finite group and \(k\geqslant 2\) be an integer. A (G, k, 1)-difference matrix (DM) is a \(k\times |G|\) matrix \(D=(d_{ij})\) with entries from G, such that for all distinct rows x and y, the multiset of differences \(\{d_{xi} d_{yi}^{-1}:1\leqslant i\leqslant |G|\}\) contains each element of G exactly once. This paper examines the existence of difference matrices with five rows over a finite abelian group. It is proved that if G is a finite abelian group and the Sylow 2-subgroup of G is trivial or noncyclic, then a (G, 5, 1)-DM exists, except for \(G \in \{{\mathbb {Z}}_3,\) \({\mathbb {Z}}_2 \oplus {\mathbb {Z}}_2,\) \({\mathbb {Z}}_4 \oplus {\mathbb {Z}}_2,\) \({\mathbb {Z}}_9\}\) and possibly for some groups whose Sylow 2-subgroup lies in \(\{{\mathbb {Z}}_2\oplus {\mathbb {Z}}_2\), \({\mathbb {Z}}_4\oplus {\mathbb {Z}}_2\), \({\mathbb {Z}}_{32} \oplus {\mathbb {Z}}_{2},\) \({\mathbb {Z}}_{16} \oplus {\mathbb {Z}}_{4}\}\), and some cyclic groups of order 9p with p prime.

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Title: Difference matrices with five rows over finite abelian groups
Authors: Rong Pan
R. Julian R. Abel
Yudhistira A. Bunjamin
Tao Feng
Tiana J. Tsang Ung
Xiaomiao Wang
Publication date: 28-01-2022
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 2/2022
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-021-00981-6

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Difference matrices with five rows over finite abelian groups

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