A note on difference matrices over non-cyclic finite abelian groups

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• 作者：Rong Pan^a ; ^b ; Yanxun Chang^a ; ^{yxchang@bjtu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Difference matrix ; Relative difference family ; Non-cyclic finite abelian group ; Finite pp-group
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 February 2016
• 年：2016
• 卷：339
• 期：2
• 页码：822-830
• 全文大小：408 K

文摘

Let (G,⋅)$(G,\cdot )$ be a finite group of order v$v$. A (G,k,λ)$(G,k,\lambda )$ difference matrix (briefly, (G,k,λ)$(G,k,\lambda )$-DM) is a k×λv$k\times \lambda v$ matrix D=(d_ij)$D=\left(d_{ij}\right)$ with entries from G$G$, so that for any distinct rows x$x$ and ddafe4740e16508bfbeb08e71" title="Click to view the MathML source">y$y$, the multiset $\{d_{xi}\cdot d_{yi}^{-1}:1\le i\le \lambda v\}$ contains each element of G$G$ exactly λ$\lambda$ times. In this paper, we are concerned about a (G,4,λ)$(G,4,\lambda )$-DM whatever structure of a finite abelian group G$G$ is. Eventually for the following two cases: (1) λ=1$\lambda =1$ and G$G$ is non-cyclic, (2) λ>1$\lambda >1$ is an odd integer, we prove that a (G,4,λ)$(G,4,\lambda )$-DM exists if and only if G$G$ has no non-trivial cyclic Sylow 2-subgroups. Moreover, we point out that a (G,4,λ)$(G,4,\lambda )$-DM always exists for any even integer λ≥2$\lambda \ge 2$ and any finite abelian group G$G$.